1. Introduction

Much attention has been paid to the subwavelength confinement of electromagnetic waves using wave-guiding techniques [1–9], for applications in high-resolution imaging and localized sensing. Here we discuss the use of a parallel-plate waveguide (PPWG) for terahertz (THz) waves, and experimentally demonstrate subwavelength confinement not only normal to the plates, as could be expected [3], but also parallel to the plates where there is no physical boundary, defying conventional wisdom in the century-old field of microwave-waveguide technology [10–17]. We achieve this complete two-dimensional (2-D) confinement by employing a finite plate-width along with a reduced plate separation. Remarkably, we find that the 2-D energy confinement within the finite-width PPWG increases exponentially as the plate separation is reduced. By adiabatically tapering the width and the separation, we focus a single-cycle pulse having a peak spectral component of 0.115 THz down to a size of 10 μm (≈λ/260) by 18 μm (≈λ/145). This corresponds to a mode area of only 2.6 × 10⁻⁵ λ², the smallest fractional mode area ever experimentally demonstrated using a wave-guiding technique at any wavelength.

This 2-D confinement is analogous to what has been observed in slot (or gap plasmon) waveguides [18,19] at optical wavelengths. These optical waveguides are geometrically similar to finite-width PPWGs, consisting of two metallic plates immersed in a homogenous dielectric medium such as air. However, the two situations are electro-dynamically distinct, since metals exhibit a strong plasmonic response at optical frequencies, but a much weaker response in the THz range. In the optical case, the 2-D confinement has been attributed to the excitation of plasmonic edge modes at the corners of the plate edges, and in particular to their mutual coupling across the gap between the two metal surfaces [18]. Our measurements of the spatial mode pattern at the output face of the PPWG suggest the relevance of these edge modes even at THz frequencies. As a result, we hypothesize that a similar mechanism mediates the observed 2-D confinement. This work opens the door to a paradigm shift in the conventional understanding of the PPWG. Furthermore, since the highly confined propagating mode exhibits no cutoff, this mode confinement suffers from no bandwidth limitations, and can therefore enable broadband subwavelength THz imaging and sensing [20].

2. Two-dimensional confinement in parallel-plate waveguides

To explore the efficiency of lateral confinement, we first study the behavior of an untapered PPWG with a plate width of w = 1 cm as in Fig. 1(a) , fabricated of two highly polished aluminum plates of length 25 cm. Using a THz time-domain spectroscopy setup, we focus a beam of single-cycle broadband THz pulses onto the input facet of the waveguide, centered on the air gap. The spot size of the input beam is chosen to have a 1/e diameter of 1 cm, matching the waveguide width. The polarization is along the y axis, perpendicular to the inner plate surfaces, to excite the TEM mode of the waveguide [21,22]. Using a fiber-coupled photoconductive receiver, we measure the spatial distribution and temporal waveforms of the radiation emerging from the output facet for several different plate separations. A 1 mm-diameter aperture held directly in front of the receiver improves the spatial resolution.

 

Fig. 1 Measurement schematic. (a) A schematic of the field measurements for untapered PPWGs. The field is directly detected using a fiber-coupled photoconductive antenna, with an aperture in front of the substrate lens to improve the spatial resolution. The antenna is oriented so as to be sensitive to the y component of the field. (b) A schematic of the measurement scheme for tapered PPWGs. In this case, an aperture-based method provides inadequate spatial resolution, so a scattering-probe technique is employed. Here, the measurement is sensitive to the z component of the field. The field can be characterized by scanning the position of the probe. The parameters describing the size of the output aperture, w_out, the plate width, and b_out, the plate separation, are shown. (c) A close-up photograph of the setup illustrated in (b).

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Figures 2(a) and 2(b) show the spatially resolved two-dimensional profiles of the transverse electric field (E _y) measured in the plane of the output facet of the PPWG with plate separations of b = 10 mm and b = 5 mm, respectively. These field maps show a surprisingly high degree of field confinement along the x axis (the unshielded direction). Moreover, there is evidence of field enhancement at the four sharp corners. This is clearer in Fig. 2(c), which shows the line profiles corresponding to the two vertical black dotted lines in Fig. 2(a). The field near the corners is ~35% larger than near the flat surfaces at x = 0. These field enhancements may be an important indication of the lateral confinement mechanism. They are probably related to plasmonic edge modes, coupled electromagnetic and electronic excitations that propagate along the metal edge. Calculations have indicated that, when two edges are in close proximity, these modes can couple to each other resulting in higher energy confinement within the plates [18]. A smaller plate separation leads to a stronger coupling, and therefore improved mode confinement.

 

Fig. 2 Characterization of the output mode in untapered PPWGs. Two-dimensional profiles of the THz electric field measured at the output facet of the 1 cm-wide PPWG with plate separation (a) b = 10 mm and (b) 5 mm. (c) Vertical line profiles along the dotted black lines in (a), showing the field enhancements near the corners. (d) Horizontal line profiles for three different plate separations, including the two shown in (a) and (b). The horizontal dashed white lines indicate the locations along which these scans have been extracted, centered between the two plates. The line profile for a 10 cm-wide PPWG is also shown for comparison (green curve).

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We can experimentally quantify the degree of mode confinement as a function of the plate separation b. Figure 2(d) shows line profiles of the measured spatial distributions along the x axis, for three different values of b. These field profiles depend strongly on the value of b. This is in contrast to the case of a much wider (10-cm) PPWG, where the field profiles are always Gaussian and independent of b [see green curve in Fig. 2(d)]. For the narrower PPWG, the departure from a Gaussian profile becomes more pronounced as b decreases. We emphasize that the curves in Fig. 2(d) are measured at the center of the air gap (i.e., at y = 0), which is as far as possible from the sharp metal corners (at y = ± b/2). Yet, for values of b less than about 5 mm, we observe a field enhancement at the waveguide edges even in the middle of the air gap. This suggests that the edge modes couple more strongly to each other as b decreases.

We define an energy confinement factor as the ratio of the total THz energy within the waveguide to the total THz energy in the plane of the output facet. This confinement factor exhibits an exponential dependence on b, converging to unity at b = 0 (see Fig. 3 ). This remarkable result indicates that the energy in the propagating mode is increasingly confined within the waveguide as b decreases.

 

Fig. 3 Mode confinement as a function of plate separation. The confinement factor as a function of the separation between the two metal surfaces at the output facet. The insets schematically illustrate how this confinement factor is defined. The curve is a fit to an exponential dependence, showing the convergence to unity at zero plate separation.

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3. Subwavelength confinement using tapered parallel-plate waveguides

With this new understanding of the crucial role of the plate separation b, we can now explore the possibility of subwavelength confinement in 2-D by adiabatically tapering the plate width and separation b [23–25]. As before, we use a waveguide of 25 cm length, but now the last 20 cm tapers uniformly from an initial width of w_in = 1 cm down to a final width, w_out, which is less than the free-space wavelength λ. We study several waveguides with different value of w_out, ranging down to 10 μm. For these waveguides, the plate separation also decreases uniformly from b_in = 1 mm (at the input end) down to a final value of b_out, which is as small as 18 μm.

In order to study the field distribution at the end of these tapered PPWGs, a 1 mm spatial resolution is inadequate. Instead of direct detection through an aperture, we employ scattering-probe imaging [9,26], which is similar to apertureless near-field microscopy [26–28]. A tapered metal probe with a point diameter of either 2 μm or 10 μm (depending on w_out) is held at the end of the waveguide, parallel to the z axis. This probe scatters the field emerging from the waveguide, where it is measured via lock-in detection [see Figs. 1(b) and 1(c)]. This technique provides subwavelength spatial resolution of the z component of the field at the probe’s apex [9].

We map E_z at the output facet of the tapered PPWGs by scanning the location of the scattering probe. Figure 4 shows typical time-domain waveforms at the output of a tapered PPWG with w_out = 10 μm and b_out = 18 μm (black and blue curves), along with an equivalent measurement on an untapered waveguide for comparison (red curve). All of these waveforms have similar temporal structure, indicating that the taper does not distort the time-domain signals. The right inset compares the amplitude spectra for the tapered and untapered cases. These are nearly identical, which demonstrates that the extreme subwavelength tapering does not impose any bandwidth restriction. It should be noted that these output spectra mimic the input spectrum, except for a broadband loss. The left inset shows a 2-D field map at the output facet of a tapered PPWG with w_out = 40 μm and b_out = 25 μm. This map, along with the waveforms, indicates a polarity reversal for E _z between the regions near the upper and lower plates of the waveguide. This polarity flip is similar to that associated with the symmetric plasmon mode of slot waveguides [23,25,29].

 

Fig. 4 Time-domain waveforms detected via scattering-probe imaging. Waveforms measured at the output facet of a tapered (w_out = 10 μm, b_out = 18 μm) and untapered waveguide. These are measured using identical methods so that the waveforms can be compared. The similarity of the red (untapered case; w_out = 1 cm) and black (tapered case) curves indicates that the waveguide taper has little effect on the shape of the time-domain signal. The right inset compares the normalized amplitude spectra, showing no bandwidth restrictions even though w_out is much smaller than the free-space wavelength in the tapered case. The left inset shows a two-dimensional field map (xy plane) at the output of a tapered waveguide (w_out = 40 μm, b_out = 25 μm) which exhibits a polarity flip between the two metal surfaces. This polarity flip, also evident in comparisons of the time-domain waveforms measured at the upper and lower plate surfaces (blue and black curves), establishes that we are observing the symmetric plasmon mode described in previous studies [23,25,29].

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By controlling both w_out and b_out, we can effectively confine this broadband THz output field to an area far below λ². Figure 5 shows line scans along the x axis, for three different PPWGs with decreasing values of w_out and b_out. In each case, the FWHM (full-width-half-max) is roughly equal in size to w_out. Since the z component is intimately coupled to the additional y component of the field via Maxwell’s equations, this is a good indication of the confinement of the resultant field at the output. This is a dramatic demonstration of the importance of plasmonic effects at THz frequencies: even though there is no material boundary along the x direction, the field is still squeezed along x to a subwavelength size. For the narrowest w_out (black dots), the x-confinement is about λ/260, corresponding to a mode area of only 2.6 × 10⁻⁵ of λ². To the best of our knowledge, this is the smallest fractional mode area ever experimentally demonstrated using a wave-guiding technique at any wavelength.

 

Fig. 5 Normalized line scans at the output of tapered PPWGs, in close proximity to the edge of the upper plate. These line scans (along the x axis, which is the unshielded direction) show the degree of localization of the THz field at the output facet of three different waveguides, with decreasing values of w_out and b_out. In each case, the smallest achievable field confinement is approximately equal to w_out, the size of the output width. The smallest width (black data points) gives the best confinement, with a FWHM of only 10 μm. Using the peak spectral component (0.115 THz) of the broadband THz pulse, this corresponds to ≈λ/260, with a mode area of 2.6 × 10⁻⁵ λ².

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4. Conclusion

In summary, we have measured the mode profile at the output of PPWGs in the THz regime. We show that it is possible to achieve confinement even parallel to the plates, defying conventional wisdom in microwave waveguide technology. (This lateral confinement is fundamentally different to what can be achieved via an index-mismatch, provided by a high-index dielectric sandwiched between the plates [30].) We apply this result to achieve extreme subwavelength 2-D confinement of THz waves in tapered PPWGs. We propose that this confinement is mediated by the coupling of plasmonic edge modes across the air gap, in analogy to the confinement mechanism for slot waveguides at optical frequencies. This result emphasizes the importance of controlling both the plate separation and the plate width in achieving subwavelength confinement, and dramatically illustrates the influence of plasmonic effects for metals at THz frequencies. Incidentally, we note that the higher energy confinement could possibly result in a higher ohmic dissipation, and thus, a higher propagation loss. Although in the THz range these losses are probably several orders of magnitude less than those at optical frequencies, this issue still remains to be examined.