Interferometric Measurement of Far Infrared Plasmons via Resonant Homodyne Mixing

We present an electrically tunable terahertz two dimensional plasmonic interferometer with an integrated detection element that down converts the terahertz fields to a DC signal. The integrated detector utilizes a resonant plasmonic homodyne mixing mechanism that measures the component of the plasma waves in-phase with an excitation field functioning as the local oscillator. Plasmonic interferometers with two independently tuned paths are studied. These devices demonstrate a means for developing a spectrometer-on-a-chip where the tuning of electrical length plays a role analogous to that of physical path length in macroscopic Fourier transform interferometers.

In this article, we demonstrate a two-dimensional (2D) plasmonic interferometer with an integrated resonant homodyne mixing element based upon a GaAs/AlGaAs HEMT with multiple gate terminals. Here it is demonstrated that biasing a gate in a HEMT near its threshold voltage while illuminated by THz radiation effectively produces a plasmonic homodyne mixing element and enables phase sensitive detection of plasma waves. When multiple plasmonic cavities are coupled to this gate-induced plasmonic mixing element, the device can be understood as a subwavelength two-path interferometer [7,22] with an integrated on-chip detector where the paths are independently tuned.
We also observe that unlike standard homodyne mixing techniques, [23] plasmonic homodyne mixing permits THz near field detection well above the conventional RC-limited bandwidth of the studied devices at their operational bias.
To describe the underlying mechanism of a solid state plasmonic interferometer, it is germane to first draw an analogy to an optical Mach-Zehnder interferometer. In Fig. 1(a), an optical Mach-Zehnder interferometer is diagrammed. Each optical path has a region of length where the permittivity ( ) and permeability ( ) of the electromagnetic medium is independently defined. If the phase velocity in these regions is given by √ ⁄ , then the phase difference of beams on these two paths is ( ). This phase difference results not from a difference in path lengths ( ), but instead from a difference in the electrical lengths of the paths ( ). When the permittivity, permeability, or both, are tunable, then so are the electrical lengths and of these regions.
A transmission line circuit representing a pair of 2D plasmonic cavities coupled to a plasmonic mixing element is shown in Fig. 1(b). This representation of a 2D plasmonic HEMT is analogous to the optical Mach-Zehnder interferometer in Fig. 1 With tuned to deplete the 2DEG below it as illustrated in Fig. 1(c), rectification takes place both at the left edge of where the signal from Path D couples to the mixing region as well as at the right edge of where the signal from Path S couples to the mixing region. Thus, the DC potential arises due to the difference between the rectified voltages on the drainside of and the source-side of . One of the underlying assumptions in this model is a loss of coherence between the two plasmonic signal channels when the mixing region of the 2DEG between them is biased near depletion. Recent modeling by Davoyan and Popov [31,32] indicates that as , the plasmon near field amplitude decays rapidly from the edges of this region into its center. This isolates the plasma excitations at opposing edges from one another, and is consistent with the experimental assumption that these decoupled plasmonic fields produce independent mixing signals. The DC measurement of the device conductance is connected with the expected THz photoresponse through Eq. 1. The factor in Eq. 1 relates the DC transport of a HEMT to its plasmonic mixing response, and is plotted in Fig. 2(b) as calculated from the conductance in Fig. 2(a). To verify that Eq. 1 and its corresponding transport measurement in Fig. 2 accurately describes the plasmonic mixing response, the photoresponse plotted in Fig where the upper and lower quantum well channels below are depleted, and also demonstrate an approximately three order of magnitude dynamic range. Although the plasmonic mixing response shown in Fig. 2 is largely unsurprising in light of the many demonstrations of this mechanism in highly varied transistor designs, material systems, and temperature ranges, [10,11,20,29,[35][36][37][38][39][40] definitively establishing the origin of this photoresponse provides the basis for describing the operation of a two-path plasmonic interferometer. Asymmetry in the plasmonic signals coupled to the mixing region, ( ) ( ) , is required for generating a non-zero photovoltage. [41,42] One means to explore introducing asymmetry into the device is by systematically voltage biasing each of the three gates.
In Fig. 3, the transport and responsivity characteristics at 8 K of the GaAs/AlGaAs HEMT in Figs. 1(c)-(e) are compared as one of the three gates is tuned independently while the other two are fixed at ground potential. The transport curves corresponding to Eq. 1 that are plotted in Fig. 3(a) are all nearly identical, consistent with the HEMT channel being homogeneous across the device and all three gates sharing an identical 2 m width. Thus, the differences in the 0.270 THz responsivity shown in Fig. 3(b) arise due to asymmetry in the device induced via the applied gate bias. The responsivity with either gate or gate , respectively, tuned is nearly identical in amplitude, but opposite in polarity. Taking to define the mixing region as illustrated in Fig. 1(d), there are two plasmonic paths feeding into this mixing region: a path formed between S and and a path formed between D and . Because these paths have different lengths, 2 m vs. 10 m, the phase and amplitude of monochromatic plasma waves impinging on the mixing region below from opposing sides is, in general, non-identical. This produces a net photoresponse because ( ) ( ) . The scenario is similar when defines the mixing region as shown in Fig. 1(e), but now the short and long plasmonic paths have exchanged relative positions in comparison to the first example.
Consistent with the measured data, this inverts the signal polarity but leaves its amplitude largely unaffected.
A third possibility, pictured in Fig. 1(c), utilizes gate to define the mixing region. In this case, the device is essentially symmetric about gate , though fabrication imperfections or misalignment of the incident radiation can introduce asymmetries. Here the photoresponse should be relatively weaker since the phase and amplitude of monochromatic plasma waves impinging on the mixing region from both paths will be nearly identical such that ( ) ( ) . In Fig. 3(b) the photoresponse with tuned has a smaller amplitude, though its measureable amplitude indicates some asymmetry in the system under THz irradiation.
Nonetheless, this is the most near to balanced configuration and also offers independent tunability of both Path D and Path S. Using this configuration, we can now systematically explore the operation of a monolithically integrated, balanced two-path plasmonic interferometer.

Two-Path Plasmonic Interferograms
With biased to deplete the 2DEG below it, the plasmonic paths between S and (Path S) and D and (Path D) may be described in terms of tunable electrical lengths. Each of these paths is 6 m long, with 2 m regions below gates and that are voltage tuned. It is these sections below gates and that are of greatest interest, and it is useful to first relate applied gate voltages to 2DEG densities. Assuming a parallel plate capacitance between each gate and the 2DEG, decreases. This is a qualitative feature of all plots in Fig. 4, and it is possible that the higher order modes in Fig. 4(d) cannot be resolved.
To further validate this proof-of-principle demonstration, we consider a second device design, shown in Fig. 5(a), where the two plasmonic paths are independently tunable four-period plasmonic crystals. [27] Here is a single 2 m gate, and and tune Path S and Path D, respectively, using four identical 2 m wide gate stripes separated by 2 m each. In this device,  Fig. 1(c) to the 18 m path lengths Fig. 5(a). The fundamental mode of the 18 m path occurs at a lower frequency than that of the 6 m path, and therefore a relatively denser set of higher order modes is anticipated for a given excitation frequency. Alternately, the coupling of four gated regions of the 2DEG in the device shown in Fig. 5(a) lifts a four-fold degeneracy, and therefore approximately four modes are expected for every one observed in the sample of Fig.   1(c). Additionally, the highest intensity signal is observed with significant tuning of gate voltage. This is analogous to observing the largest signal in Fig. 4 at any electrical lengths but the smallest measured. One possibility consistent with a recent study of localized modes in terahertz plasmonic crystals [27] is that specific modes in the spectrum couple less well to the mixing region as well as to the THz excitation field due to their confinement adjacent to contact S or D. Although the distributed nature of the THz excitation precludes validation of this hypothesis using a lumped source to model the plasmonic near field amplitude, the nonmonotonic behavior of signal intensities is suggestive of the localization of plasmon modes in Path S and Path D.

Conclusions
We have demonstrated an approach to integrate on-chip plasmonic interferometry with a widely-used plasmonic detection technique in this article. Although an antenna provided the distributed excitation of the signal channels and the LO of the plasmonic mixer, the presented approach should readily transfer to waveguide-coupled structures [7,22] if the LO and signal channels are suitably isolated. The phase relationship between the LO and signal channels is determined by the coupling of the antenna excitation to HEMT terminals. Isolation of these channels would allow for control of their relative phase and potentially a quadrature measurement to extract both the amplitude and phase of an incident THz signal. This possibility arises because the plasmonic mixer is a field rather than power detector. While intensity interferograms are often measured by bringing two paths coincident upon a power detector, here field phase information is partially preserved by independently generating a DC signal from each path and reading out to a single differential channel.
The sensitivity of 2D plasma excitations to their environment portends intriguing possibilities for sensor development. Though the presented devices based on GaAs/AlGaAs heterostructures need both cryogenic cooling and a vacuum environment to operate, other plasmonic materials such as graphene have neither as a requirement. The electromagnetic screening of 2D plasma waves by a metal terminal is a limiting case of environment modifying plasmon dispersion. However, more subtle effects, particularly in graphene, can arise due to plasmon-phonon coupling with an adjacent material [44,45] or the coupling of plasmons with an adsorbed polymer [46]. The research in this area thus far has focused on optical techniques. The devices presented in this article, however, suggest that an electro-optical approach to graphene near-field plasmonic sensing might also be fruitful.
Integration of interferometric elements into a voltage-tunable microelectronic plasmonic device provides potential advantages over a Fourier transform interferometer, particularly in the far infrared. Though the substantial reduction in optical path length is beneficial, the most significant advantage is provided by the broad voltage tunability. Optical interferometers must mechanically tune a path over lengths on the order of meters to measure high resolution spectra.
In the studied low dimensional plasmonic interferometers, the electrical length can comfortably tune over an order of magnitude, though the intrinsic plasmonic losses ultimately limit resolution. Despite this limitation, this study provides an important step towards device and system integration of multiple signal path plasmonic detectors for future generations of far infrared sensing technologies. x ' t x t p C g p y t x MX. (c) A scanning electron micrograph of a two-path plasmonic interferometer where gate G2 of a HEMT defines the mixing element and Path S and Path D are tuned by G1 and G3, respectively. The gates are all approximately 2 m wide and separated by 2 m. The distance between the Ohmic contacts S and D is 14 m. In (d) and (e) the same device is shown but with G1 and G3, respectively, defining the mixing region.