Free-Space Subterahertz-Field Polarization Controlled by Selection of Waveguide Modes

We experimentally study the free-space electromagnetic field emitted from a multimode rectangular waveguide equipped with a diagonal-horn antenna. Using the frequency range of 215–580 GHz, a photomixer is used to launch a free-space circularly polarized electromagnetic field, exciting multiple modes at the input of the rectangular waveguide via an input diagonal-horn antenna. A second photomixer is used, together with a silicon mirror Fresnel scatterer, to act as a polarization-sensitive coherent detector to characterize the emitted field. We find that the radiated field, excited by the fundamental waveguide mode, is characterized by a linear polarization. In addition, we find that the polarization of the radiated field rotates by ${45}^{\circ }$ if selectively exciting higher-order modes in the waveguide. Despite the higher-order modes, the radiated field appears to maintain a predominant Gaussian beam character, since an unidirectional coupling to a detector was possible, whereas the unidirectionality is independent of the frequency. We discuss a possible application of this finding.

Figure 1

(a) Conventional way of transforming a left-circularly polarized electromagnetic field into a linearly polarized field, by rotating mechanically a quarter-wave plate to the desired angle. (b) Multimode waveguide and diagonal-horn antenna assembly, acting as a nonmechanical ${45}^{\circ }$ polarization rotator, demonstrated in this work. The excitation of multiple waveguide modes with the circularly polarized field distributes the field energy equally among the waveguide modes, indicated by the red and blue photons. (c) Exciting only the fundamental waveguide mode leads to a radiated field from the output diagonal-horn antenna with a well-defined linear polarization along the $\eta$ axis (copolarization axis) of the aperture coordinate system of the diagonal-horn antenna, shown as the red photon. Furthermore, exciting higher-order waveguide modes leads to a radiated field with a ${45}^{\circ }$ rotated polarization due to a superposition of linear polarizations along the $\eta$ axis and $\xi$ axis (crosspolarization axis), shown as the blue photon.

Figure 2

Near-field simulation results of the electric portion of the electromagnetic field, expressed as electric field intensity, in the experimentally implemented antenna-feed waveguide (rectangular section with dimensions $a$ and $b$) and bell-mouthed diagonal-horn antenna shape, obtained with the cst software [27]. In the antenna-feed waveguide, which is further connected to the waveguide channel [see Figs. 3, 3, and 3], a certain mode of the electric portion of the electromagnetic field is excited predominantly in the $H$ or $E$ plane. After the waveguide excitation, however, an electric field develops in both planes of the diagonal-horn antenna. The electric field intensities in the diagonal-horn antenna planes are mode dependent. The figure summarizes the results for the first three propagating modes at reasonably selected frequencies: (a) ${\mathrm{TE}}_{10}$ (270 GHz), (b) ${\mathrm{TE}}_{20}$ (480 GHz), and (c) ${\mathrm{TE}}_{01}$ (480 GHz). Note that the overall values of the electric field intensities, the intensities being quantified by the scales in units of V/m, in the $E$ and $H$ planes of the diagonal-horn antenna for the ${\mathrm{TE}}_{10}$ mode (a) are approximately swapped between same $E$ and $H$ planes for the ${\mathrm{TE}}_{01}$ mode (c). The overall values of the electric field intensities in the $E$ and $H$ planes of the diagonal-horn antenna are approximately the same for the ${\mathrm{TE}}_{20}$ mode (b). With these three modes in superposition, polarization rotation by ${45}^{\circ }$ occurs around the wavevector $k_z$, suggested by Fig. 3.

Figure 3

Design, components, and assembly of the fabricated diagonal-horn antenna and waveguide device. (a) The completed assembly, diagonal-horn antennas opened along the $E$ plane (b) that are connected by a full-height rectangular waveguide (c). In the left-hand diagram of (b), the black profile emphasizes the antenna profile; the right-hand diagram of (b) shows the subdivision into cross sections (i)–(iv), shown in more detail in (e). (d) Cross sections of the diagonal-horn antenna aperture (left) and the profile (right). (e) Cross sections indicated in (b), showing the inner conductor shape of the waveguide and the diagonal-horn antenna, with the electromagnetic field of the fundamental ${\mathrm{TE}}_{10}$ mode in the waveguide and diagonal-horn antenna at positions i and iv. The $\eta$-$\xi$ aperture coordinate system is shown in iv.

Figure 4

Measurement system to measure the polarization at the output of the diagonal-horn antenna. Coherent sub-THz source and detector are used and the lock-in detected signal is displayed as a function of frequency. The planar silicon mirror (PM; TYDEX [32]) acts as a Fresnel scatterer and enables the polarization measurement. It reflects linearly polarized electric fields with polarization components parallel and perpendicular to the plane of incidence (paper plane), with opposite signs (and slightly different magnitudes) into the aperture of the coherent detector. Two successive measurements result in the detector currents $I_{\mathrm{dc}}^E$ and $I_{\mathrm{dc}}^H$, Eqs. (3a) and (3b), when the $E$ and $H$ planes of the waveguide (see Fig. 3) and diagonal-horn antenna are successively aligned parallel to the plane of incidence. This implements a direct measurement of the correlation function in Eq. (4). A beam absorber (BA) attenuates a standing wave in the setup.

Figure 5

Overview of the results on the measured polarization of the rectangular waveguide and diagonal-horn antenna assembly over the frequency range 215–580 GHz. The line at the top indicates the presence of a number of higher-order modes. Note the scale change at 315 GHz. (a) Experimental (black) and simulation (red) results together with results for the single-mode (orange) and multimode (blue) model. The experimental data points (black) are evaluated from histograms, such as that shown in the inset, by dividing the positive counts by the negative ones, pos/neg (left $y$ axis). The black histograms contain the experimental values of the instantaneous phase-phase correlation function for the output electromagnetic field, $\cos ({\varphi }^E)\times \cos ({\varphi }^H)$ [Fig. 4 and Eq. (4)]. Panels (b) and (c) suggest a change of the direction of the polarization of the diagonal horn output field due to multiple mode excitation of the waveguide. Panel (b) suggests the polarization direction when only the fundamental mode is excited in the waveguide and panel (c) suggests the polarization direction when multiple modes are excited.

Figure 6

Detected multimode sub-THz field scattering off the ${45}^{\circ }$-tilted silicon mirror. (a) Measured detector response roll-off, expressed by $\mid {\mathcal{I}}_{\mathrm{dc}}\mid$, Eq. (2). The black and the red traces show measurements when the $H$ or $E$ plane of the diagonal-horn antenna and waveguide are aligned parallel to the planar silicon-mirror plane of incidence. Note that the fluctuations of $\mid {\mathcal{I}}_{\mathrm{dc}}\mid$ are due to the planar silicon-mirror transmission and reflection and partly also due to standing waves, but are not caused by the noise of the detector. The dashed line shows the measured Lorentzian detector response with a roll-off time constant of 500 fs, determined by a transmission measurement between the source and detector only. (b) Phase of the detected sub-THz field within a selected frequency range [dotted lines in (a)]. Because of the detected multimode field, it shows phase jumps compared to the more ordered phase as a function of frequency in Fig. 7 where only a single mode is detected. The data in the shown frequency window in (b) are employed to determine the data point at 450 GHz in Fig. 5 of the main text.

Figure 7

Detected single-mode sub-THz field scattering off the ${45}^{\circ }$-tilted silicon mirror. (a) Measured Fresnel magnitude of a detected free-space sub-THz field, being excited by the waveguide ${\mathrm{TE}}_{10}$ mode and radiated into free space by the diagonal-horn antenna. (b) Measured instantaneous phase ${\varphi }^{H,E}$, after scattering off the planar silicon mirror. In (a) the smooth thick lines are obtained using Fresnel theory and are compared with the experimental data (thin lines). We show the measured phase in (b) only within a selected frequency range [dashed region in (a)] for reasons of clarity. With the $E$ plane of the waveguide and diagonal horn aligned parallel to the planar silicon-mirror plane of incidence (as shown in Fig. 4 of the main text), the scattering magnitude $\mid r_{\parallel }\mid$ and the phase ${\varphi }^E$ are measured. Aligning the waveguide and diagonal-horn $H$ plane parallel to the planar silicon-mirror plane of incidence measures the scattering magnitude $\mid r_{\perp }\mid$ and the phase ${\varphi }^H$. Note that in (a) the experimental data are rescaled to the magnitude of the theoretical Fresnel magnitudes that our experiment determine up to a constant factor. Furthermore, we added a factor $\mathcal{A}=0.85$ to the experimental $\parallel$ component to match the theory, as reasoned in Appendix pp2.

Figure 8

Electromagnetic far-field simulations of the diagonal-horn antenna at a reference distance of 1 m from the aperture, performed with the cst software [27]. The images show the absolute value of the electric component of the far-field radiation pattern in the Ludwig-3 coordinate system [34]. The first three propagating modes emitted by the diagonal-horn antenna are shown at reasonably selected frequencies. The white dashed line indicates a cut through the map, shown on top of each figure. Panels (a), (c), and (e) show the copolarization component $\mid {\mathcal{E}}_{\eta }\mid$ and panels (b), (d), and (f) show the crosspolarization component $\mid {\mathcal{E}}_{\xi }\mid$; see the definition of the $\eta$-$\xi$ aperture coordinate system in Fig. 1 of the main text. The far-field intensities of the co- and crosspolarization components for the ${\mathrm{TE}}_{10}$ mode are approximately interchanged for the $\mathrm{T}{\mathrm{E}}_{01}$ mode, and the total absolute electric field in the co- and crosspolarization components for the $\mathrm{T}{\mathrm{E}}_{20}$ mode is approximately the same. This indicates that, for the superposition of the three shown propagating modes, the polarization is rotated by ${45}^{\circ }$ around the wavevector $k_z$.

Figure 9

Crosspolarization amplitude content ${\mathcal{E}}_{\mathrm{crpol}}$ (a) and the phase difference between the co- and crosspolarization ($⟨{\phi }_{\mathrm{cpol}}{⟩}_{\varphi ,\theta }-⟨{\phi }_{\mathrm{crpol}}{⟩}_{\varphi ,\theta }$) components (b) for the ${\mathrm{TE}}_{10}$ mode. (c) Cross- and copolarization amplitude content (${\mathcal{E}}_{\mathrm{crpol}}$ and ${\mathcal{E}}_{\mathrm{cpol}}$) for the ${\mathrm{TE}}_{10}$ mode and higher-order modes for selected frequencies (black $\widehat{=}{\mathrm{TE}}_{10}$, red $\widehat{=}{\mathrm{TE}}_{20}$, blue $\widehat{=}{\mathrm{TE}}_{01}$, orange $\widehat{=}{\mathrm{TE}}_{11}$, and green $\widehat{=}{\mathrm{TM}}_{11}$). (d) Phase difference between the co- and crosspolarization components for the modes in (c), showing that the radiated polarization components in each mode are hardly phase shifted with respect to each other. The figure shows results only at selected frequencies, because of the numerically time consuming simulations. When the first three modes (black, red, and blue) propagate for frequencies larger than 400 GHz, a nearly balanced amount of co- and crosspolarization amplitude components are indicative of a polarization rotation by ${45}^{\circ }$. The other two higher-order modes, orange and green, contribute with equal parts to the co- and crosspolarizations and, hence, further stabilize the polarization rotation by ${45}^{\circ }$.

Figure 10

Analogy between photon-pair production during parametric down-conversion in a nonlinear crystal (a) and due to charge-light coupling in a mesoscopic superconducting two-photon device (b) as realized in Refs. [15, 16, 18, 19] (green, red, and blue parts of the figure). In (b), the red and blue photons excite the waveguide (for instance, through chip-waveguide coupling [37]) and are radiated in the same or in different polarization states, dependent on their frequencies, into the detector and trigger apertures after scattering off a frequency-selective beamsplitter. The ADCs count coincidences $N_{\mathrm{coinc}}$ between the detector and trigger and photon counts $N_2$ of the trigger alone. For detector characterization, the thermal photon population of the environment, $⟨F^{†}F⟩$, has to be considered.