Directional Coupling to a λ/5000 Nanowaveguide

Abstract

Silicon-based microdevices are considered promising candidates for consolidating several terahertz technologies into a common and practical platform. The practicality stems from the relatively low loss, device compactness, ease of fabrication, and wide range of available passive and active functionalities. Nevertheless, typical device footprints are limited by diffraction to several hundreds of micrometers, which hinders emerging nanoscale applications at terahertz frequencies. While metallic gap modes provide nanoscale terahertz confinement, efficiently coupling to them is difficult. Here, we present and experimentally demonstrate a strategy for efficiently interfacing subterahertz radiation (λ = 1 mm) to a waveguide formed by a nanogap, etched in a gold film, that is 200 nm (λ/5000) wide and up to 4.5 mm long. The design principle relies on phase matching dielectric and nanogap waveguide modes, resulting in efficient directional coupling between them when they are placed side-by-side. Broadband far-field terahertz transmission experiments through the dielectric waveguide reveal a transmission dip near the designed wavelength due to resonant coupling. Near-field measurements on the surface of the gold layer confirm that such a dip is accompanied by a transfer of power to the nanogap, with an estimated coupling efficiency of ∼10%. Our approach efficiently interfaces millimeter waves with nanoscale waveguides in a tailored and controllable manner, with important implications for on-chip nanospectroscopy, telecommunications, and quantum technologies.

Keywords: nanophotonics; near-field imaging; plasmonics; terahertz photonics; terahertz time domain spectroscopy.

Figure 1

Concept schematic of the terahertz directional nanocoupler presented here. (a) An x-polarized broadband terahertz pulse propagates in z through a substrateless silicon waveguide (SiWG). At λ ∼ 1 mm, it efficiently couples to the metallic gap mode over a length L, due to phase matching with the fundamental mode of the nanogap waveguide (NGWG) above it. (b) Cross section schematic of the coupling region in the xy plane, showing geometry and material distribution (light blue: quartz; dark blue: silicon; yellow: gold; white: air). Here, t = 50 μm, h = 100 nm, H = 250 μm, W = 190 μm, w = 200 nm, and s is varied. Note that the schematic is not to scale to simplify labeling and that waveguide and gap widths differ by 3 orders of magnitude: w ∼ λ/5000. (c) Calculated dispersion of of the fundamental modes of the individual SiWG (red) and NGWG (blue), as a function of frequency, highlighting the phase matching point near 0.3 THz.

Figure 2

(a) Real part of the effective index of relevant modes as a function of the gap width w, at f = 0.29 THz and s = 60 μm, with all other parameters as per Figure 1 caption. The dielectric WG (SiWG) remains unchanged with w (red dashed line). The effective index of the nanogap mode increases with decreasing w (blue dashed line), and the phase matches with the WG mode at w = 200 nm. The modes hybridize when the waveguides are adjacent, resulting in two supermodes (SMs, solid lines), whose effective indices split at the phase-matching point. (b) Corresponding imaginary part of the effective index. (c, d) Colormap of the WG and gap modes (z-component of the Poynting vector), with a detail of the white dashed region in the inset of (d). (e, f) Hybrid SM₁ and SM₂ that emerge when the waveguides couple, respectively. A logarithmic scale is used due to the large relative field intensity in the nanogap. Window size in (c)–(f): 600 μm × 600 μm. Scale bar in (d) inset: 400 nm.

Figure 3

Overview of the effective index dispersion, transmission spectra, and power exchange properties for different separations s. (a) For larger separations (s = 150 μm, blue box), the real part of the effective index for both isolated and coupled WG/gap modes cross, and (b) the imaginary parts anticross. (c) Associated calculated power in the dielectric waveguide P_Si as a function of frequency for L = 5 mm (red line, left axis on dB scale), showing resonant transmission dip accompanied by a corresponding transmission peak in the nanogap waveguide power P_NG (blue line, right axis). At the resonance frequency (0.29 THz, black arrow), (d) shows that the P_Si monotonically decreases with increasing length L, with P_NG plateauing at ∼0.05, decreasing at longer lengths due to losses. Bringing the waveguides closer (s = 10 μm, green box) leads to (e) an anticrossing in the real parts of the SM dispersions, (f) and a crossing of their imaginary parts, due to stronger coupling. (g) The resulting transmission dip is sharper for shorter device lengths (here: L = 1.0 mm) due to directional coupling (red lines, left axis). Correspondingly, more power is coupled to the plasmonic gap. Note that the P_Si minimum has shifted to 0.32 THz (black arrow in (g)). (h) At this frequency, the power in the SiWG oscillates and decays (red line), whereas the relative power in the NGWG peaks at L ∼ 1.0 mm.

Figure 4

Summary of the far-field transmission experiments. (a) Side view of the experiments. An x-polarized THz pulse (red) is coupled to the fundamental mode of the substrateless silicon waveguide (SiWG, dark blue). A series of metal nanogap waveguides (yellow) on a quartz substrate (light blue) are moved in x for a given separation s. (b) Top view of the experiment. The NGWGs vary in length (here: L = 3.7–4.5 mm). We measured the transmitted intensity |E_T|² of the THz pulse through the SiWG as a function of x. (c) Microscope images of the experiment for different values of L as labeled. The green is a custom holder for the SiWG. Note the moving NGWGs for a fixed input condition. (d) Optical micrograph of the SiWG and microbridges. (e) Optical micrograph of the gold NGWGs deposited on quartz surrounded by a 100 μm trench, included to facilitate visual inspection. Inset: scanning electron micrograph of the NGWG edge, showing the 200 nm gap. (f) Measured colormap of the transmitted intensity through the SiWG as a function of frequency and the x-position of the NGWG sample for a constant s ∼ 50 μm, covering both the trench and the 4.5 mm NGWG sample.

Figure 5

Dependence of the nanocoupler transmission on the length L of the NGWG and separation s to the SiWG. (a) Measured transmitted intensity spectrum when the SiWG and NGWG are in near-contact, as shown in Figure 4(c), for different L’s. (b) Eigenmode method simulations of the power spectrum in the SiWG when s = 30 μm. (c) Measured transmitted intensity spectrum as a function of s for L = 4.1 mm. Note the emergence of a resonance as the waveguides are brought closer. (d) Corresponding eigenmode method simulations of the power spectrum in the dielectric waveguide as a function of s.

Figure 6

Experimental demonstration of terahertz radiation coupling to a λ/5000 nanogap waveguide. (a) Microscope image of a terahertz near-field (NF) antenna scanning the surface of a quartz substrate supporting the NGWGs, aligned with a SiWG. Note that the incoming electric field is x-polarized and that the NF antenna detects x-polarized fields. (b) Optical micrograph of the substrate, highlighting that the nanogap waveguide is in the center of the SiWG. The spots are residuals from the fabrication process and represent minor surface contaminations that do not provide additional scattering; see Animation 1 in the Supporting Information. (c) Far-field transmission: the red trace shows the transmitted |E|² (i.e., the far-field transmission) through the silicon WG for the sample configuration shown in (a) and (b). Note the resonance at 0.31 THz, which is absent for the bare SiWG (blue trace). (d) Snapshots of the measured electric field at (i) t = 0 ps (no field). The horizontal lines mark the nominal SiWG boundary. The vertical dashed lines highlight the edges of the NGWG (L = 4.4 mm). The orange horizontal line shows the nominal location of the NGWG, as per (b). (ii) Field snapshot at t = 27 ps (main pulse is from the scattered field that does not couple to the SiWG), (iii) t = 127 ps (evanescent field of the NGWG mode), and (iv) t = 169 ps (field scattered by the end of the NGWG). The complete animation showing pulse propagation is reported in Animation 1 in the Supporting Information. (e) Measured dispersion of the detected NGWG mode using the data in (d)(iii). (f) Measured distribution of the scattered electric field intensity at the end of the NGWG using the data in (d)(iv). The scattered intensity is centered at 0.31 THz, confirming that the dip in (c) is due to coupling to the nanogap. Inset: comparison of the measured total intensity scattered by the nanogap (dashed box in (d)(iv)) and the estimated intensity exiting the SiWG from two independent measurements, showing percentage-level coupling efficiency. Also shown below the curve is an example of measured intensity scattered by the nanogap following a temporally gated Fourier transform for the case of 0.31 THz.