We experimentally demonstrate complete two-dimensional (2-D) confinement of terahertz (THz) energy in finite-width parallel-plate waveguides, defying conventional wisdom in the century-old field of microwave waveguide technology. We find that the degree of energy confinement increases exponentially with decreasing plate separation. We propose that this 2-D confinement is mediated by the mutual coupling of plasmonic edge modes, analogous to that observed in slot waveguides at optical wavelengths. By adiabatically tapering the width and the separation, we focus THz waves down to a size of 10 μm (≈λ/260) by 18 μm (≈λ/145), which corresponds to a mode area of only 2.6 × 10−5 λ2.
©2010 Optical Society of America
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 , 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 . 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 .
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.
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 . A smaller plate separation leads to a stronger coupling, and therefore improved mode confinement.
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.
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 win = 1 cm down to a final width, wout, which is less than the free-space wavelength λ. We study several waveguides with different value of wout, ranging down to 10 μm. For these waveguides, the plate separation also decreases uniformly from bin = 1 mm (at the input end) down to a final value of bout, 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 wout) 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 .
We map Ez 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 wout = 10 μm and bout = 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 wout = 40 μm and bout = 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].
By controlling both wout and bout, 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 wout and bout. In each case, the FWHM (full-width-half-max) is roughly equal in size to wout. 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 wout (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.
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 .) 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.
This work was funded in part by the National Science Foundation and by the Air Force Office of Scientific Research through the CONTACT program.
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