Terahertz frequency counter based on a semiconductor-superlattice harmonic mixer with four-octave measurable bandwidth and 16-digit precision

Figures 1(a) and (b) illustrate the schematic and block diagram of the dual-counter frequency difference (DCFD) method. Its concept has been adopted for the characterization of various frequency counters in the THz to visible-light region [13, 35, 36]. The DCFD method is composed of one CW–THz source and two independent SLHM-based THz counters. The THz radiation emitted from the CW–THz source at ν_THz is split into two beams. They are introduced into the horn antennas of SLHM1 and SLHM2, respectively. They can downconvert ν_THz to the intermediate RF-beat signals at ${f}_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{t}i}\left(i=1,2\right)$ by heterodyne-mixing with the nearest k_i th harmonics of microwave LO signals. The LO signals are produced by a common RF synthesizer1 with a frequency f_RF1 through independent frequency multipliers with the multiplication factors m_i ; the LO frequency becomes m_i f_RF1. Since the SLHM behaves as a THz-comb generator whose mode spacing is m_i f_RF1, f_beati in figure 1(b) is written as

$$\begin{equation}{f}_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{t}i}={\nu }_{\mathrm{T}\mathrm{H}\mathrm{z}}-{k}_{i}\enspace {m}_{i}\enspace {f}_{\text{RF}1}+{n}_{\mathrm{t}\mathrm{o}\mathrm{t}i},\end{equation} \tag{ 1 }$$

where n_toti is the sum of the equivalent frequency noise originating from the shot noise (n_shoti ), thermal noise of SLHM (n_thi ) and electronic noise of amplifier (n_ei ). The difference between two beat frequencies, Δf_beat, is given by

$$\begin{equation}{\Delta}{f}_{\text{beat}}\equiv {f}_{\text{beat}2}-{f}_{\text{beat}1}={n}_{\text{tot}2}-{n}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{1}},\end{equation} \tag{ 2 }$$

which represents canceling out the instabilities of ν_THz and f_RF1 as common-mode noise when k₁ m₁ = k₂ m₂. Accordingly, the DCFD method can reveal the intrinsic noise of the SLHM-based THz counter, which consists of quantum, thermal and technical noises. One practical drawback of this method is the paucity of THz sources oscillating at above 1 THz with the sufficient output power for the precise frequency measurement.

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Figures 1(c) and (d) shows the schematic and block diagram of the phase-locked THz–QCL (locked-QCL) method. This method is comprised of a CW–THz source phase-locked to a stable THz frequency reference and one SLHM-based THz counter. The THz–QCL is a promising CW–THz source for counter characterization above 1 THz owing to the prominent milliwatt-order output power; other CW–THz sources emit lower power in typically tens of microwatts [1]. However, the THz–QCLs often have a large frequency jitter of several MHz, which is caused by the mechanical vibration from a Stirling refrigerator; it is necessary to cool the THz–QCL to cryogenic temperature to achieve the population inversion between the two inter-subbands. In the locked-QCL method, this jitter noise is suppressed to a nonsignificant level for the absolute frequency measurement of the THz–QCL.

To decrease the miscounting rate in the precise measurement of f_beat4 yielded by SLHM2, the free-running THz–QCL at ν_QCL must be stabilized to the k₁th LO harmonic at k₁ m₁ f_RF1 produced in SLHM1. Their heterodyne beat signal at f_beat3 is phase-locked to an output signal at f_RF2 from RF synthesizer2. The control signal extracted by using a mixer is fed back to the THz–QCL to modulate ν_QCL after appropriate filtering and amplification in the phase-locked loop (PLL) servo. The resultant frequency ${\nu }_{\mathrm{T}\mathrm{H}\mathrm{z}}^{\prime }$ in figure 1(d) is calculated as

$$\begin{align}{\nu }_{\text{THz}}^{\prime }& =\frac{{G}_{\text{op}}}{1+{G}_{\text{op}}}\left\{\frac{{\nu }_{\text{QCL}}}{{G}_{\text{op}}}+\left({k}_{1}{m}_{1}{f}_{\text{RF}1}+{f}_{\text{RF}2}\right)\right.\\ & \quad -\left.\left({n}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{1}}+{n}_{\mathrm{d}1}+\frac{{v}_{\text{h}1}}{{D}_{1}}\right)\right\},\end{align} \tag{ 3 }$$

where G_op ≡ D₁ H₁ is an open-loop gain of the PLL, and D₁ and H₁ represent a frequency–voltage and a voltage–frequency converter, respectively. n_d1 is an equivalent frequency noise attributable to D₁, and v_h1 is an input noise voltage of H₁. Using equation (3) on the assumption of G_op ≫ 1 and k₁ m₁ = k₂ m₂, f_beat4 can be written as

$$\begin{equation}{f}_{\text{beat}4}\approx \frac{{\nu }_{\text{QCL}}}{{G}_{\text{op}}}+{f}_{\text{RF}2}+{n}_{\text{tot}2}-{n}_{\text{tot}1}^{\prime },\end{equation} \tag{ 4 }$$

where ${n}_{\text{tot}1}^{\prime }\equiv {n}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{1}}+{n}_{\mathrm{d}1}+{v}_{\text{h}1}/{D}_{1}$. The first term on the right side of equation (4) indicates the suppression of frequency fluctuation from THz–QCL by the PLL gain; consequently, the contribution to the f_beat4 measurement is negligible. The second term, which represents a phase-locked beat frequency, is also insignificant because the fluctuation of f_RF2 is at least three orders of magnitude smaller than those of the LO harmonics. As a result, the fluctuation of f_beat4 is dominated by ${n}_{\text{tot}2}-{n}_{\text{tot}1}^{\prime }$ in equation (4), and then the evaluation of f_beat4 can be regarded as almost equivalent to measuring Δf_beat by the DCFD method. The locked-QCL method serves to verify the measurement accuracy of the SLHM-based THz counter above 1 THz, where the powerful THz–QCLs are available.

Figure 2 shows the experimental setup for the DCFD method. Two measured ν_THz by the inline and perpendicular-THz counters were compared to obtain Δf_beat on the computer. The CW–THz sources employed are frequency multipliers (Virginia diodes, AMC303 and WR2.8 × 3R5) with low-noise RF synthesizer0 (Agilent, E8251A) and a frequency-multiplied Gunn oscillator (Radiometer Physics GmbH) phase-locking to a reference frequency from the synthesizer. The frequency multipliers extend the microwave signal from the synthesizer0 into the THz radiation ranging from 88 GHz to 360 GHz. On the other hand, the Gunn oscillator can provide THz signals from 618 GHz to 654 GHz. Their output powers measured at the specific frequencies are summarized in table 1.

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The radiation emitted from the CW–THz sources was collimated by a Tsurupica lens and divided into two orthogonal directions by a high-resistivity float zone silicon (HRFZ-Si) beam splitter with 1 mm thickness. Each radiation was focused by a lens to maximize the coupling efficiency to the SLHM through a diagonal horn antenna. Apart from the horn for THz-signal input, the SLHM device has a microwave LO input port and an intermediate frequency output port [32]. The microwave LO signal is provided to the SLED via a WR-06 rectangular waveguide to convert the LO power into the harmonics current in the THz band. The LO signals were generated from another metrology-grade RF synthesizer1 (Agilent, E8247C) using two independent amplifier/multiplier chains with a multiplication factor m of 12 (Virginia Diode, WR6.5AMC-I). The LO frequency can be tuned from 110 GHz to 170 GHz with a typical output power of 16 mW. The intermediate beat frequencies at f_beat1 and f_beat2 from a coaxial connector of each SLHM were set to be 180 MHz. Each beat signal was amplified by a multistage RF amplifier (RFA) with a power gain of 60 dB and a noise figure of 2.9 dB (mini-circuits). After transmitting tunable-bandpass filters with 5% bandwidth of the center frequency for the signal conditioning (TBPF1, TBPF2), f_beat1 and f_beat2 were measured by independent Π-type RF counters (Pendulum, CNT-91). The beat signal measured had to attain the SNR of more than 25 dB with a 100 kHz resolution bandwidth (RBW) to prevent the RF counter from miscounting the beat frequency. Two RF counters were synchronously operated in the external arming mode by a common square-wave trigger signal at 1 Hz with 99% duty cycle. Such operation allows the conservation of the phase coherence between the two beat signals registered; the common-mode noise can be rejected from Δf through data analysis. Although there exists the measurement dead time of 10 ms during 1 s gate time for every frequency counting, the instability of the measured signal in the Allan deviation is degraded by only 1% in the case of the white phase noise [37]. The RF synthesizers and counters employed here were externally referenced to a hydrogen maser frequency standard (Anritsu, RH401A).

The experimental setup for the locked-QCL method is depicted in figure 3. Although the configuration of the perpendicular-THz counter was identical to that for the DCFD method, the inline system was modified to be engaged in the PLL for the THz–QCL. Two THz–QCLs were employed for this method. One was a third-order distributed-feedback laser with an output power of 150 μW at ν_QCL of 2020 GHz. The other was a Fabry–Perot laser with 2 mW output power at 2828 GHz. Both oscillated at single frequencies with linear polarization and TEM₀₀ mode (Longwave Photonics). These lasers were placed in a vacuum chamber and cooled to 50 K using a Stirling refrigerator. The beam irradiated from the THz–QCL was focused by a silicon-meniscus lens and subsequently separated into two beams by an HRFZ-Si beam splitter. The inline beam was received by a diagonal horn of the SLHM1 to transfer to its SLED. The ν_QCL was downconverted to f_beat3 by mixing with the appropriate kth harmonic of the microwave LO signal. The microwave signal was the frequency-multiplied output from RF synthesizer1. Table 1 summarizes the output frequencies from RF synthesizer1 and the multiplication factor m as well as the harmonic numbers k. To obtain an error signal for the PLL, f_beat3 was compared with an output signal at 125 MHz from RF synthesizer2 using a double-balanced mixer (DBM). The error signal was filter-amplified by PLL-servo electronics and afterward directly added to the laser current via a resistive summer to shift the laser frequency from ν_QCL to ${\nu }_{\text{THz}}^{\prime }$. Its control bandwidth was about 5 MHz, which was restricted by the phase delay caused by the servo electronics. The typical continuous phase-locking time was within several hours, which arose from the reversed sign of the error signal during the operation [30].

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Figure 4 plots the typical RF spectra of f_beat2 and f_beat4 obtained by the perpendicular-THz counters in the DCFD and locked-QCL methods. The spectra of f_beat1 for the inline-THz counter in the DCFD method and f_beat3 as an inloop signal of the PLL in the locked-QCL method showed similar shapes to the related spectra in figure 4. The narrow spectra of f_beat2 regarding the frequency multipliers with the RF synthesizer0 in figures 4(a)–(c) reflected its low phase noise, while the broad ones associated with the Gunn oscillator in figures 4(d) and (e) were originated from its intrinsically wide linewidth. The sidelobes next to f_beat4 at 125 MHz in figure 4(g) were generated by the slightly distorted signal at the RFA1 output. The measured beat signals had SNR of more than 26.4 dB with a 100 kHz RBW as summarized in table 1. This satisfies the empirically imposed requirement on the beat SNR for precisely counting the frequencies by RF counters. The observed linewidth of 1 Hz for each beat spectrum was limited by the RBW of the RF spectrum analyzer. In particular, such a narrow linewidth of f_beat4 indicated that the PLL suppressed well the jitter noise on ν_QCL.

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The Allan deviations for ν_THz from 120 GHz to 654 GHz, which were measured by the inline and perpendicular-THz counters, are plotted with triangles and squares in figures 5(a2)–(e2). Two instabilities at each ν_THz were equivalent and restricted by the instabilities of RF synthesizers used in the experiment, as represented by dash-dot lines. The instabilities corresponding to one THz counter are plotted with circles. They were obtained from Δf_beat for each ν_THz, which eliminated the instabilities of RF synthesizers as the common-mode noise. The THz-counter instabilities had a 1/τ dependence and dropped to 1 × 10⁻¹⁶ within the averaging time τ of 10 000 s. To clarify the noise source dominating the instabilities, we evaluated the equivalent frequency instability ${\sigma }_{y}^{(\mathrm{a})}$ of the RFA as a function of the beat SNR; a sinusoidal test signal at 180 MHz was amplified by the RFA with changing the SNR of its output signal between 30 dB and 41 dB, then the measured instability of the output signal with each SNR derived ${\sigma }_{y}^{(\mathrm{a})}=\left(-2.6{\times}1{0}^{-6}\enspace \mathrm{S}\mathrm{N}\mathrm{R}+4.9{\times}1{0}^{-2}\right)/\tau \enspace {\nu }_{\mathrm{T}\mathrm{H}\mathrm{z}}$. The dashed lines in figure 5 represent the RFA instabilities calculated from the measured SNR of f_beat1 and f_beat2. They were in good agreement with the instabilities of the THz counter, whereas the influences of other noise sources were estimated to be negligible at present. The measurement limit ${\sigma }_{y}^{(\mathrm{c})}$ of the RF counter was determined to be ${\sigma }_{y}^{(\mathrm{c})}=5.2{\times}1{0}^{-3}/\tau \enspace {\nu }_{\mathrm{T}\mathrm{H}\mathrm{z}}$, which was also lower than the THz-counter instabilities. It should be noted that only 15% power of the whole 120 GHz radiation detected was coupled to the SLED through the horn antenna owing to its higher cut-off frequency; the coaxial cable for the intermediate-frequency output behaved as a more efficient antenna for the 120 GHz-signal detection. Nevertheless, the SLHM worked as a fundamental frequency mixer with a low instability at 120 GHz.

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The relative instabilities of phase-locked THz–QCLs to the LO harmonics as the THz frequency reference are plotted by upside down triangles in figures 5(f2) and (g2). They achieved a lower instability than the RFA noise level as represented by the dashed lines; this occurs in the feedback loop, where the RFA noise is suppressed together with the ν_QCL fluctuation. The genuine instabilities of the THz–QCLs are shown by circles. They were computed from the frequency time series in figures 5(f1) and (g1), which were measured by the perpendicular-THz counter; the missing data at early times are due to the exclusion of frequency time series during unlocked states of the PLL, where the recorded frequency largely deviates from the correct one. As explained in section 2.2, the genuine instabilities of the THz–QCLs correspond to the instability of the THz counter. They plunged into the 10⁻¹⁷ level within 2000 s. The excursions from a 1/τ dependence were accompanied by the cycle slips in the PLL and the f_beat4 miscountings owing to the time-varying low SNR. The instability of the THz counter between 2020 GHz and 2828 GHz was also limited by the RFA noise. The THz-counter instability can be improved substantially by increasing the amplitude of the beat signal or by decreasing the electronic noise from the RFAs. Such further improvements will reveal the intrinsic noise of SLED and lead to a better understanding of the miniband electron transport in the semiconductor superlattice. The Allan deviations of the THz counter at 1 s averaging time are summarized in figure 6(a). Accordingly, the instability of the SLHM-based THz counter developed was found to be restricted by the RFA noise at the wide THz band between 120 GHz and 2.8 THz.

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The mean of the fractional frequency difference and the standard error at each ν_THz were computed to be less than 4.6 × 10⁻¹⁷ and below 9.6 × 10⁻¹⁷, respectively (figure 6(b)). They were obtained from 30 data sets, where each data set comprised of 1000 data points with a sampling rate of 1 Hz. The statistical error was regarded as the standard error, although the instability of the THz counter indicated not random processes but phase-coherent process with white phase noise, which has a 1/τ dependence in the Allan deviation. No systematic frequency offset were confirmed within the standard error except for the results at 654 GHz and 2020 GHz; we attribute their slight offset to the unbalanced noise sidebands around the carrier RF-beat signals, which enforce a deviation from the correct ν_THz. Even though there exists such an offset at the specific ν_THz, the SLHM-based THz counter, which is referenced to a future optical SI second with the uncertainty of 10⁻¹⁸ level [38], is capable of determining the absolute THz frequency with an uncertainty of less than 1 × 10⁻¹⁶ over the four-octave THz range. This is because the potential measurement limit of the THz counter below 1 THz, which was revealed to have the above stated uncertainty by the DCFD method, can be achieved beyond the present limit by employing an LO signal generated from such future SI-traceable reference with enough accuracy and stability instead of the RF synthesizer1; consequently the absolute frequency measurement with the stated uncertainty. Similarly, the locked-QCL method has verified the comparable potential limit of the THz counter over 1 THz by the effective rejection of common noise mainly associated with the RF synthesizer1, as explained in section 2.2. It is worth mentioning that the relatively large uncertainty at 120 GHz as shown in figure 6(b), which restricts the overall measurement uncertainty of the THz counter, is attributed to not only the lower ν_THz affecting the dominating RFA instability ${\sigma }_{y}^{(\mathrm{a})}$ but also the lower beat SNR due to the inferior coupling efficiency of 120 GHz radiation into the SLED through the horn antenna. Owing to the paucity of suitable CW–THz sources, thorough research on the wide measurable spectrum of the developed THz counter was difficult to be carried out. This sparsely left several uninvestigated bands, which can be covered with third and some even LO harmonics as depicted in figure 6(c). The presence of those LO harmonics is reasonable in accordance with the previous reports regarding the harmonics generation by the SLED [39]. Therefore, the THz counter developed is expected to work in the broad THz band with measurement uncertainties comparable to the ν_THz investigated.

Table 2 shows the uncertainty budget of the THz counter developed. The total uncertainty of 6.9 × 10⁻¹⁷ except for the contribution from the LO-phase noise is the potential measurement limit and agrees well with the measured uncertainty of the developed system. The uncertainty of the electric noise caused by the RFA is the main contribution to the potential measurement limit. This uncertainty is owing to the f_beat measurement with the practical beat SNR after the signal amplification. The RF-counting standard error represents the statistical uncertainty arose from the frequency measurement using the RF counter, that is related to its measurement limit ${\sigma }_{y}^{(\mathrm{c})}$. The PLL servo error gives the possibility of the deviation of the THz–QCL frequency from the reference LO harmonic, which was evaluated from f_beat3 in the PLL for the THz–QCL; this contributes to only the locked-QCL method. The LO-phase noise represents the phase noise of the frequency-multiplied output of the RF synthesizer1, and thus the uncertainty originated from its phase noise is adopted here; the uncertainty of the hydrogen maser as the external LO reference is not included in that of the LO-phase noise. Each uncertainty for the RFA electric noise, RF-counting standard error, PLL servo error and LO-phase noise was computed from independently acquired 30 data sets, where each data set was composed of 1000 data points with 1 Hz sampling. The estimated uncertainties for the thermal and shot noise of the SLHM are negligibly small contributions (<1 × 10⁻²⁰). Note that these uncertainties depend on the averaging time because of the random effect. No inherent frequency offset is expected for all uncertainty sources listed.

Apart from the measurement at 120 GHz, even LO harmonics were utilized for the other ν_THz measurements with the various THz sources, as summarized in table 1. This is because the higher SNR of the heterodyne beat signal was always achieved by mixing with the even harmonics rather than the adjacent odd ones. Further study of the amplitude of LO harmonics generated in the SLHM is worth pursuing to determine the upper limit of the measurable spectrum. The power of the kth LO harmonics ${P}_{\text{HM}}^{(k)}$ relative to that of the THz radiation ${P}_{\text{THz}}^{\prime }$ coupled into the horn antenna of the SLHM can be obtained from the following equation:

$$\begin{equation}{P}_{\text{HM}}^{(k)}={A}_{0}\enspace \frac{{P}_{\text{noise}}^{2}}{{P}_{\text{THz}}^{\prime }}\enspace {\mathrm{S}\mathrm{N}\mathrm{R}}_{k}^{2},\end{equation} \tag{ 5 }$$

where SNR_k is the measured SNR of a beat signal between the kth harmonics and the THz source, P_noise is the total noise power that appeared in the beat measurement; actually, P_noise can be replaced with the electronic noise power of the RFA in our experiment. A₀ is assumed as a constant coefficient for the sake of simplification. Note that ${P}_{\text{THz}}^{\prime }$ is not the original power from a THz source in table 1 but the compensated power for the path loss in the air due to water absorption and the optics, and the coupling efficiency that is defined by a ratio of the effective aperture area of the horn antenna to the spot size of the THz beam on the aperture. Figure 7(a) plots ${P}_{\text{HM}}^{(\mathrm{k})}$ as a function of the LO harmonic number k. The LO power was changed from 0.71 mW to 16 mW for each measurement in order to maximize the SNR_k . The overlapped spectral coverage of the adjacent LO harmonics enabled their power measurement with the identical THz sources. Nevertheless, the harmonics contained in the gray area have no information because of the lack of proper THz sources oscillating near the corresponding harmonics frequencies. Obviously, the power of even harmonics exceeds those of adjacent odd harmonics. This implies that the conversion efficiency of even harmonics from the LO signal was superior to that of odd ones. Such a power distribution could be caused by the Esaki–Tsu current–voltage characteristic with antisymmetry: the induced current in the SLED by the LO voltage forms a rectangular wave with bumps in the positive and negative amplitudes, which are originated from the negative differential conductivity [25, 40, 41]. The envelope of the even harmonics' power had a k^−3.4 dependence up to k = 20, which was almost consistent with the characteristics of the SLED-based frequency multipliers [31, 32]. The power of the higher harmonics, however, declined more steeply with a k⁻³⁷ dependence. The harmonics with k > 23 were difficult to observe under our experimental condition. Thus, the maximum possible frequency to be measured by our THz counter is approximately 3.7 THz when the highest 22th harmonic is generated from the LO signal at 170 GHz. Further extension of the measurable range up to 4.7 THz is expected by the cryogenic cooling of the SLHM because of the reduced scattering of miniband electrons by acoustic and polar optical phonons, consequently increasing the electron drift velocity in the semiconductor superlattice [42].

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Figure 7(b) shows the ${P}_{\text{THz}}^{\prime }$ required for 25 dB SNR of the beat signal with 100 kHz RBW; this SNR must be satisfied to prevent f_beat from being miscounted by conventional RF counters. The required ${P}_{\text{THz}}^{\prime }$ against the sets of even or odd LO harmonics is given by

$$\begin{equation}{P}_{\text{THz}}^{\prime }=\begin{cases}\mathrm{exp}(-6.3)\enspace {k}^{3.4}\quad & \quad (k\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\enspace \cap \enspace 2{\leqslant}k{\leqslant}20)\\ \mathrm{exp}(-106.8)\enspace {k}^{37}\quad & \quad (k\enspace \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\enspace \cap \enspace k{ >}20)\\ \mathrm{exp}(-0.3)\enspace {k}^{2.9}\quad & \quad (k\enspace \mathrm{o}\mathrm{d}\mathrm{d}\enspace \cap \enspace 3{\leqslant}k{\leqslant}21).\end{cases}\end{equation} \tag{ 6 }$$

According to equation (6), a CW–THz source oscillating at 3.7 THz must have an output power of more than 1.9 mW for the frequency measurement by our THz counter.