Self-calibration technique for characterization of integrated THz waveguides

Emerging high-frequency accelerator technology in the terahertz regime is promising for the development of compact high-brightness accelerators and high resolution–power beam diagnostics. One resounding challenge when scaling to higher frequencies and to smaller structures is the proportional scaling of tolerances which can hinder the overall performance of the structure. Consequently, characterizing these structures is essential for nominal operation. Here, we present a novel and simple self-calibration technique to characterize the dispersion relation of integrated hollow THz waveguides. The developed model is verified in simulation by extracting dispersion characteristics of a standard waveguide a priori known by theory. The extracted phase velocity is in good agreement with the true value. In experiments, the method demonstrates its ability to measure dispersion characteristics of nonstandard waveguides embedded with their couplers with an accuracy due to systematic errors of $\approx 0.5\%$. Equipped with dielectric lining, the metallic waveguides act as slow wave structures, and the dispersion curves are compared without and with dielectric. A phase synchronous mode, suitable for transverse deflection, is found at 275 GHz.

Figure 1

(a) Cross section of a dielectric loaded waveguide with a inner radius $a$, outer radius $b$, and permittivity ${\epsilon }_r$. The dielectric is illustrated in gray, while the surrounding metallic wall is copper colored. (b)–(c) Schematic of the reflected wave’s phase shift on the reference plane when moving an obstacle through the waveguide in sub-wavelength steps.

Figure 2

Error Network model, summarizing the couplers and the free space as $P$-network and describing the obstacle as $Q$-network. The DUT represents the waveguide which imposes $S_{11}=S_{22}=0$.

Figure 3

Schematic of the experimental setup and the network representation of the components. The horn couplers form the $P$-network, while the obstacle is represented by the $Q$-network. The embedded waveguide is the DUT with the unknown $S$-parameters. Not to scale.

Figure 4

Out- and in-coupling horns considered as single network device with simulated scattering parameters.

Figure 5

Real and imaginary parts of the simulated reflection coefficient $S_{11}(l)$ as a function of the obstacle position, at a fixed frequency of 303.16 GHz based on the WR3.4 waveguide as DUT and an arbitrary generated obstacle reflection. The asymmetry of $\Re S_{11}$ reveals the difference compared to a sine shape.

Figure 6

Phase velocity $v_{\mathrm{ph}}$ (blue line) extracted from the simulation using the four-term error model and its relative deviation $\delta v_{\mathrm{ph}}$ (orange line) from the predicted known phase velocity for the WR3.4 waveguide.

Figure 7

Experimental implementation of the schematic setup of Fig. 3. (1) Extender with waveguide port (2) standard gain horn antenna (3) mounted DUT (structure (c) from Fig. 8) (4) inserted obstacle mounted on a movable stage (stage not shown).

Figure 8

Metallic waveguides under test. They are not lined with dielectric. (a) Copper structure with drilled channel. (b) Same design but machined in brass. (c) Split-block structure fabricated by wire EDM. (d)–(f) Microscope images of the apertures with measured radii $a$ of $511\pm 10$, $505\pm 7$, and $657\pm 7\mu \mathrm{m}$, respectively.

Figure 9

Measured phase velocity dispersion (blue) and fit with the analytical model (orange, dashed) of the drilled copper waveguide, shown in Fig. 8. The prediction (green) with uncertainty (green band) is based on radius measurements from Fig. 8, including errors in circle detection and finite pixel resolution. The measurement does not fall into the $1\sigma$-uncertainty band of the prediction. Uncertainties in the measurement are too small to be visible on the chosen axis scale. A few selected points of the measurement are highlighted as bullets for improved visualization with respect to the fit.

Figure 10

Measured (blue line), fitted (orange, dashed) and predicted phase velocity dispersion of the brass waveguide, shown in Fig. 8. Prediction based on Fig. 8.

Figure 11

Measurement of split-block waveguide analyzed by the four-term error model, Eq. (5), and prediction.

Figure 12

Multimode analysis based on the filter diagonalization method, together with expected analytical higher order mode dispersions. ${\mathrm{TE}}_{01}$ and ${\mathrm{TM}}_{11}$ have identical dispersion relations.

Figure 13

Comparing copper block waveguide with additional oxide layer. The frequency range is truncated to 290 GHz due to very low signal above this threshold, as seen already partially by the ripples.

Figure 14

Real part of the measured reflection coefficient $S_{11}$ depending on the obstacle position of the oxidized waveguide. The fit model is extended with a nonvanishing attenuation coefficient $\alpha$. The envelope is based on $\beta =0$. Note that the generator is to the right side of the plot which is why the attenuation appears as an exponential growth.

Figure 15

Phase velocity dispersion of a dielectric loaded waveguide using a 3D printed capillary. Inlet: cross sectional view of the waveguide aperture.