Characterizing the properties (e.g., effective dielectric constant *ε*_{eff}, attenuation constant *α*, and characteristic impedance *Z*_{0}) of terahertz (THz) superconducting transmission lines is of particular interest in designing on-chip integrated THz bandpass filters, which are a critical component for THz astronomical instruments, such as multi-color camera and broadband imaging spectrometers. Here, we propose a novel method for the characterization of three parameters (*ε*_{eff}, *α*, and *Z*_{0}) of THz superconducting transmission lines. This method measures the ratio of the THz signal powers through two different-length branches of the superconducting transmission line to be measured. In addition, only one measurement is required for an all-in-one device chip, including an antenna, a half-power divider, the superconducting transmission line to be measured, and two detectors. The key point is that the superconducting transmission line to be measured is impedance-mismatched with the two integrated detectors. The method is validated through simulation and measurement for superconducting coplanar waveguide transmission lines around 400 GHz.

## I. INTRODUCTION

Observing the optically invisible universe at terahertz (THz) wavelengths, loosely defined as 0.1–10 THz, will revolutionize many areas of astronomy, such as the star formation and galaxy evolution, as nearly half of the photon energy in the universe after the cosmic microwave background (CMB) falls within this frequency regime.^{1} In addition, the CMB polarization measurement at THz frequencies (up to hundreds of GHz) is currently a frontier in the detection of the primordial gravitation wave (PGW).^{2} For space-/air-borne and ground-based THz telescopes, multicolor cameras, broadband imaging spectrometers, and multibeam heterodyne receivers incorporated with superconducting detectors are the major choices of instrumentation. On-chip THz bandpass filters, made from superconducting planar transmission lines (e.g., coplanar waveguides and microstrips), are a critical component in developing the instruments mentioned above, particularly emerging on-chip broadband imaging spectrometers.^{3,4} It is thus of particular importance to measure precisely the properties of superconducting planar transmission lines at THz frequencies, including the effective dielectric constant *ε*_{eff}, attenuation constant *α*, and characteristic impedance *Z*_{0}.

Characterization of THz superconducting planar transmission lines is quite challenging as it needs to be carried out at low temperatures and THz frequencies. Newly developed measurement methods adopt a common strategy that uses a THz source through quasi-optical irradiation and on-chip integrated THz bolometric detectors. One method is through the measurement of a planar Fabry–Pérot interferometer/resonator made of a long superconducting transmission line, which is followed by an integrated bolometric detector.^{5,6} The effective dielectric constant *ε*_{eff} of the long superconducting transmission line is extracted from the interval between the resonance harmonics around the frequency of interest, while the attenuation constant *α* is extracted from the internal quality factor *Q*_{i} of the resonator that needs to know the coupling quality factor *Q*_{c} in advance, generally through simulation. Another method is by comparing the power transmissions measured for two superconducting transmission lines with different lengths.^{7–9} It requires knowing the characteristic impedance *Z*_{0} in advance (through simulation, for example) and two separate measurements, i.e., one DUT (device under test) for the measurement of *α* with the two superconducting transmission lines being perfectly impedance-matched with respective detectors and the other DUT for the measurement of *ε*_{eff} with one superconducting transmission line impedance-mismatched with the corresponding detector. Fabricating this kind of DUT for the measurement of *α* is not an easy task, given the fact that the actual *Z*_{0} might differ from the simulated one at THz frequencies. Note that for the two methods introduced above, the characteristic impedance *Z*_{0} cannot be measured.

Here, we propose a novel method for the measurement of three parameters (*ε*_{eff}, *α*, and *Z*_{0}) of THz superconducting transmission lines. Only a single DUT chip, as shown in Fig. 1(a), is required for this measurement, with the superconducting transmission line to be measured (two branches of different lengths) being impedance-mismatched with the two integrated detectors. Obviously, the mismatching requirement is much less stringent for fabrication. The DUT chip is composed of a planar THz antenna, a planar half-power divider, two different-length branches of the superconducting transmission line to be measured, followed by respective THz bolometric detectors with their impedances differing from the characteristic impedance of the THz superconducting transmission line to be measured. Here, the two output ports of the half-power divider are symmetrical and are connected to the superconducting transmission lines. Note that the antenna and the two detectors can be designed at the frequencies of interest and the impedances of the two detectors can differ so long as they are different from the characteristic impedance of the superconducting transmission line to be measured. The three parameters (*ε*_{eff}, *α*, and *Z*_{0}) of the superconducting transmission line are extracted from the ratio of the measured powers from the two detectors. Note that the symmetry of the 3 dB power divider should be guaranteed to have a high accuracy for the parameter extraction.

## II. METHODOLOGY

*Z*

_{1},

*Z*

_{2}) and (

*P*

_{1}′,

*P*

_{2}′), we can have the THz power ratio of

*P*

_{1}′/

*P*

_{2}′ written as

*Z*

_{i}(

*i*= 1 or 2) is equal to

*Z*

_{0}(

*Z*

_{li}+

*Z*

_{0}

*tanh γl*

_{i})/(

*Z*

_{0}+

*Z*

_{li}

*tanh γl*

_{i}), where

*γ*=

*α*+

*jβ*is the propagation constant of the superconducting transmission line, with the phase constant

*β*given by 2

*πf*(

*ε*

_{eff})

^{0.5}/

*c*, in which

*f*is the frequency and

*c*is the speed of light. Since the two output ports of the power divider are symmetrical, the THz power ratio of

*P*

_{1}′/

*P*

_{2}′ is independent of the broadband performance of the power divider. As shown in Fig. 1(a),

*Z*

_{li}and

*l*

_{i}represent the detector impedances and the superconducting-transmission-line lengths, respectively. The THz signal powers received by the two detectors can be given by

^{10}

_{i}= (

*Z*

_{li}−

*Z*

_{0})/(

*Z*

_{li}+

*Z*

_{0}). Thus, the ratio of the THz signal powers received by the two detectors can be expressed as

*β*and then the effective dielectric constant

*ε*

_{eff}. Furthermore, given different lengths of the two superconducting transmission lines (

*l*

_{1}and

*l*

_{2}) and different impedances of the two detectors (

*Z*

_{l1}and

*Z*

_{l2}), one can extract the attenuation constant

*α*and characteristic impedance

*Z*

_{0}of the superconducting transmission line by fitting the measured THz-signal-power ratio of

*P*

_{1}/

*P*

_{2}.

## III. RESULTS AND DISCUSSION

^{11}to validate this method. For a circuit shown in Fig. 1(a), we assumed the superconducting transmission lines to be measured of (

*ε*

_{eff},

*α*, and

*Z*

_{0}) as (6.25, 0.01 dB/mm, and 40 Ω) and of respective lengths as

*l*

_{1}= 1 mm and

*l*

_{2}= 8 mm. The port impedances of the two detectors and the planar antenna were assumed to be

*Z*

_{l1}= 100 Ω,

*Z*

_{l2}= 150 Ω, and

*Z*

_{l3}= 80 Ω. Figure 1(b) shows the simulated THz-signal-power ratio of

*P*

_{1}/

*P*

_{2}in the frequency range from 300 to 500 GHz. It can be clearly seen that there are two standing waves of different frequency periods (marked by two dashed boxes), corresponding to the two branches of the superconducting transmission line to be measured. The effective dielectric constant

*ε*

_{eff}of the superconducting transmission line is related to the frequency periods through

*f*

_{i}is the frequency period of the standing waves. Using Eq. (3), we fitted the simulated result shown in Fig. 1(b) based on the nonlinear least squares method and found that the three fitted parameters of (

*ε*

_{eff},

*α*, and

*Z*

_{0}) are consistent with the designated values. Note that the initial value of the effective dielectric constant used for the fit is given by Eq. (4).

As we know, a mismatched antenna can result in standing waves and affect the received power of the detector. To avoid the calibration of the input power at the antenna port and to reduce systematic errors, the power ratio of *P*_{1}/*P*_{2} is adopted in this method. By changing the antenna-port impedance *Z*_{l3}, we re-simulated the designated circuit shown in Fig. 1(a). It has been found that while both *P*_{1} and *P*_{2} vary significantly, the power ratio remains unchanged. Hence, the influence of the mismatched antenna can be eliminated by using the power ratio of *P*_{1}/*P*_{2}. We also did simulation by changing the attenuation constant and characteristic impedance of the THz superconducting transmission line. As plotted in Figs. 1(c) and 1(d), the change in *P*_{1}/*P*_{2} is significant. Thus, this method is effective also for extracting the parameters of *α* and *Z*_{0}.

Second, we designed and fabricated two devices to verify the proposed measurement method. As shown in Fig. 2, here, we choose a superconducting CPW transmission line for the ease of fabrication. The only difference between the two devices is that air bridges—across the measured superconducting CPW transmission line—are adopted for one device (D2) but not for the other (D1). Note that air bridges may suppress the slot-line mode existing in CPW transmission lines, thus reducing the transmission loss.^{12} Both devices integrate a double-slot antenna, a half-power divider with its three ports with air bridges across the CPW grounds, two branches of different lengths of the superconducting CPW transmission line under measurement, and two titanium (Ti) hot-electron bolometric (HEB) detectors (both with a designated normal-state impedance of 130 Ω). The superconducting CPW transmission line has a 2-*µ*m gap and a 3-μm-wide central conductor, corresponding to a characteristic impedance of 50 Ω (simulated by ADS) by assuming perfect conductor—different from the detectors’ impedances for the sake of mismatching, and the measured lengths of its two branches are *l*_{1} = 0.223 mm and *l*_{2} = 10.231 mm. The two devices are fabricated from a single-layer Nb superconducting film (∼100 nm thick), which is electron-beam evaporated on a 320-*µ*m thick silicon substrate with a resistivity above 10^{4} Ω$\xb7$cm. In addition, the air bridges, with an interval of 45 *µ*m and a width of 5 *µ*m, are also made of Nb and fabricated by an etching process.^{13} In addition, we examined the symmetry of the 3 dB power divider and found that the difference in central-conductor width between the two CPW channels was nearly negligible (less than 0.1 *µ*m). An independent measurement on the 3 dB power divider shows that the THz-power ratio between the two channels is less than 0.13 dB, mainly due to the measurement error.

Figure 2(c) depicts the schematic diagram of a dual-channel setup for this verification experiment. The device under measurement is glued to an elliptical silicon lens for coupling the input THz signal from a 325–500 GHz Schottky multiplier chain (THz source) and is then mounted in a copper block anchored onto the 3-K cold stage of a pulse-tube cryocooler. The integrated two Ti HEB detectors based on Johnson noise readout,^{14} with a typical NEP of 5 × 10^{−10} W/Hz^{0.5} at 3 K, are followed by respective low-noise amplifiers (0.1–14 GHz, also at 3 K) and a shared room-temperature readout circuit (with an IF bandpass of 3.9–4.5 GHz). The two bandpass filters (BPFs) in the readout circuit are for better suppression of interference signals, and the square-law detector of 1 mV/*μ*W responsivity is used to rectify the noise power of the Ti HEB detectors. In addition, the function generator shown in Fig. 3(c) is utilized to provide Joule heating power to calibrate the THz power absorbed by the two Ti HEB detectors.^{14} Note that the impedance of the Ti HEB detector in the terahertz frequency band is equal to its normal-state impedance.^{15} In this case, the normal-state impedance of the Ti HEB detector is measured by the four-wire method. The measured impedances of the Ti HEB detectors (Ch1 and Ch2) are 126.1 and 130.5 Ω for the device D1 and 134.1 and 135.9 Ω for the device D2, respectively. The difference in the impedances of the Ti HEB detectors is mainly due to the fabrication process. Figure 3(a) shows the powers received by the Ti HEB detectors (Ch1 and Ch2) for the two devices (D1 and D2) when we sweep the THz source from 325 to 500 GHz. It can be clearly seen that they vary significantly with frequency owing mainly to the frequency dependence of the input power of the THz source, standing wave in the optical paths, and the frequency selectivity of the double-slot antenna. When we plotted the power ratio of *P*_{1}/*P*_{2}, as shown in Fig. 3(b), those effects were almost disappeared and periodic fluctuations due to mismatching between the measured superconducting CPW transmission line and the two detectors could be clearly seen.

To obtain the effective dielectric constant *ε*_{eff}, the attenuation constant *α*, and the characteristic impedance *Z*_{0} of the measured superconducting CPW transmission lines, we fit the measured power ratios in the frequency range from 390 to 420 GHz (with an interval of 0.1 GHz) in terms of Eq. (3). The fitting is based on the nonlinear least squares method with 5 and 4 iterations for the two devices (D1 and D2), respectively. Note that the three parameters (*ε*_{eff}, *α*, and *Z*_{0}) are assumed to be frequency independent. As shown in Fig. 4, the measured and fitted results are in good agreement. The fitted values of the three parameters (*ε*_{eff}, *α*, and *Z*_{0}) are given in Table I with a confidence level of 95%. The effective dielectric constant *ε*_{eff} appears larger than the simulated one (6.17 by ADS) by assuming perfect conductor and *ε*_{r} = 11.7 as the dielectric constant of the silicon substrate. The reason is that superconducting material will introduce kinetic inductance^{16} at low temperatures, thus resulting in an increase in phase constant of the CPW transmission line. The characteristic impedance *Z*_{0} is also larger due to the same effect. By comparing the results for D1 and D2, we can find that the effective dielectric constant *ε*_{eff} is almost unchanged, while the attenuation constant *α* is reduced from 0.163 to 0.067 dB/mm, indicating that air bridges do play a role in balancing the ground potential of the superconducting CPW transmission line and suppressing the slot-line mode. The characteristic impedance *Z*_{0} of the superconducting CPW transmission line with air bridges is smaller, which might be related to the total length of the air bridges and needs to be further studied.

## IV. CONCLUSIONS

In conclusion, we have proposed a novel method for the measurement of three parameters (*ε*_{eff}, *α*, and *Z*_{0}) of THz superconducting transmission lines with an all-in-one device chip with two integrated detectors impedance-mismatched with the superconducting transmission line to be measured. The method is first validated through ADS simulation and then by measurement for superconducting CPW transmission line (with and without air bridges) around 400 GHz. It has been found that this method is effective particularly in eliminating system errors, such as calibration and optical-path alignment. It is applicable to different types of THz superconducting transmission lines.

## ACKNOWLEDGMENTS

This work was supported, in part, by NSFC under Grant Nos. 11925304 and 12020101002 and, in part, by CAS under Grant Nos. YJKYYQ20170031 and GJJSTD20210002.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Zhaohang Peng**: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Bowen Fan**: Data curation (equal); Investigation (supporting); Software (lead); Validation (equal); Writing – review & editing (supporting). **Wei Miao**: Validation (equal); Writing – review & editing (supporting). **Zheng Wang**: Resources (lead). **Yuan Ren**: Supervision (equal). **Jing Li**: Validation (equal). **Shengcai Shi**: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.