Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser

A single, free-running, dual-wavelength mode-locked, erbium-doped fibre laser was exploited to measure the absolute frequency of continuous-wave terahertz (CW-THz) radiation in real time using dual THz combs of photo-carriers (dual PC-THz combs). Two independent mode-locked laser beams with different wavelengths and different repetition frequencies were generated from this laser and were used to generate dual PC-THz combs having different frequency spacings in photoconductive antennae. Based on the dual PC-THz combs, the absolute frequency of CW-THz radiation was determined with a relative precision of 1.2 × 10−9 and a relative accuracy of 1.4 × 10−9 at a sampling rate of 100 Hz. Real-time determination of the absolute frequency of CW-THz radiation varying over a few tens of GHz was also demonstrated. Use of a single dual-wavelength mode-locked fibre laser, in place of dual mode-locked lasers, greatly reduced the size, complexity, and cost of the measurement system while maintaining the real-time capability and high measurement precision.


Principle of Operation
THz-comb-referenced frequency measurement is based on heterodyne photoconductive mixing between CW-THz radiation and a PC-THz comb 6,7 . Two essential conditions must be satisfied: (1) a PCA must work as a broadband heterodyne receiver with high sensitivity for THz radiation at room temperature, and (2) the generated PC-THz comb should cover the whole THz band. When CW-THz radiation (freq. = f THz ) is photoconductively mixed with one mode of a single PC-THz comb (freq. interval = f rep , comb mode nearest in frequency to f THz = m), f THz is given by

THz r ep beat
where f beat is the beat frequency between CW-THz radiation and the m-th comb mode. Next we consider the photoconductive mixing of CW-THz radiation with dual PC-THz combs having different frequency spacings (PC-THz comb 1, freq. interval = f rep1 , comb mode nearest in frequency to f THz = m; PC-THz comb 2, freq. interval = f rep2 , comb mode nearest in frequency to f THz = m). In this case, when f rep2 > f rep1 , f THz is given by where f beat1 is the beat frequency between the CW-THz radiation and the m-th comb mode in PC-THz comb1, and f beat2 is the beat frequency between the CW-THz radiation and the m-th mode in PC-THz comb2. From Eq.
(2), m can be calculated by where the signs of f beat1 and f beat2 are determined by the relative positions of f THz , mf rep1 , and mf rep2 . Figure 1 shows the relative position of f THz (see green line) to the nearest modes mf rep1 (see red lines) and mf rep2 (see blue lines) in the dual PC-THz combs, where (a) f THz < mf rep1 < mf rep2 , (b) mf rep1 < f THz < mf rep2 , and (c) mf rep1 < mf rep2 < f THz . Since the frequency difference between f rep1 and f rep2 ( = f rep2 − f rep1 = ∆f rep ) is the denominator of Eq. (3), a highly stable frequency difference is essential for accurately determining m. The relative positions of f THz , mf rep1 , and mf rep2 can be determined from the simultaneous measurements of f rep1 , f rep2 , f beat1 , and f beat2 as follows: Therefore, m can be obtained by (a) f THz < mf rep1 < mf rep2 , (b) mf rep1 < f THz < mf rep2 , and (c) mf rep1 < mf rep2 < f THz .

Results
Free-running, dual-λ mode-locked Er:fibre laser. Figure 2(a) shows the configuration of the free-running, dual-λ mode-locked Er:fibre laser oscillator. With birefringence-induced filtering and loss control effects 12,13,19 , in addition to the adjustment of the polarization state in the ring cavity, simultaneous mode-locking centred on the 1530-nm and 1560-nm regions can be realized. As shown in Fig. 2 Fig. 2(c). In order to meet the optical power and pulse duration requirements for PCAs, the λ 1 and λ 2 pulses from the laser oscillator were separated by a coarse-wavelength-division-multiplexing bandpass filter (CWDM-BPF) [not shown in Fig. 2(a)]. Figure 3(a and b) show optical spectra and RF spectra of the λ 1 and λ 2 pulses after passing through the CWDM bandpass filter. The dual-λ mode-locked fibre laser light was successfully separated into each component in the optical region and the RF region. Then, the components were amplified and spectrally broadened by erbium-doped fibre amplifiers (EDFAs) and the following SMF, respectively. As shown in Fig. 3(c), the optical spectrum of the amplified λ 1 pulsed light covered the whole C band, whereas that of the amplified λ 2 pulsed light was located at the shorter wavelength side. The mean power and the pulse duration were 27 mW and 130 fs for the amplified λ 1 pulsed light [see Before performing real-time measurement of f THz with dual PC-THz combs, we investigated the frequency characteristics of this free-running laser. We first measured the temporal fluctuations of f rep1 and f rep2 with a frequency counter (Agilent 53132 A). Figure 4(a) shows the fluctuations with respect to different gate times. Due to the free-running operation without active frequency control, the fluctuations of f rep1 and f rep2 did not decrease over a gate time of 0.1 s. However, these fluctuations were comparable to those of other commercialized, free-running single-wavelength lasers 10,11 ; this is clear evidence that the two mode-locked operations at λ 1 and λ 2 do not compete with each other and are completely independent of each other. Figure 4(b) shows the temporal fluctuations of f rep1 and f rep2 , where their frequency deviations from the initial values are indicated by δf rep1 and δf rep2 . A slow drift was clearly confirmed for both, indicating changes in the environmental conditions in the fibre cavity. However, it should be emphasized that the temporal behaviours of δf rep1 and δf rep2 were the same. This is because the λ 1 and λ 2 pulses co-propagated in the same ring cavity and experienced similar disturbances. As a result of such common-mode behaviour of δf rep1 and δf rep2 , ∆f rep was highly stable, as shown in Fig. 4(c). The mean and standard deviation of ∆f rep in Fig. 4(c) were 1764.97 Hz and 0.24 Hz, respectively. Such high stability of ∆f rep was useful for the correct determination of m and f THz based on Eqs. (4 to 6). Therefore, even though f rep1 and f rep2 were not actively stabilized, this dual-λ mode-locked fibre laser can be used for measuring f THz in real time and with high-precision using dual PC-THz combs. Figure 5 shows a schematic diagram of the setup for measuring the frequency of CW-THz radiation, consisting of three main parts. The first part is the laser source, including a free-running, dual-λ mode-locked Er:fibre laser oscillator and two EDFAs. The second part is composed of the optical and THz systems for frequency measurement of CW-THz radiation, a CW-THz test source, a pair of low-temperature-grown (LT) InGaAs/InAlAs PCAs (PCA1 and PCA2), and their affiliated components. The third part is the data acquisition electronics.

Real-time determination of f THz with dual PC-THz combs.
The amplified λ 1 pulsed light at f rep1 from one EDFA (EDFA1) was used for generating a PC-THz comb in PCA1 (PC-THz comb 1, freq. spacing = f rep1 ), whereas the amplified λ 2 pulsed light at f rep2 from another EDFA (EDFA2) was used for generating a PC-THz comb in PCA2 (PC-THz comb 2, freq. spacing = f rep2 ). When the CW-THz radiation was incident on both PCA1 and PCA2, photoconductive mixing between the CW-THz radiation and the dual PC-THz combs and the following electronic processing resulted in the generation of beat signals with frequencies f beat1 and f beat2 . On the other hand, RF signals related to f rep1 or f rep2 (freq. = 30f rep1 − f LO and 30f rep2 − f LO ) were obtained by the photodetectors (PD) and subsequent electric heterodyning with a local oscillator (LO, freq. = f LO ). Temporal waveforms of f beat1 , f beat2 , 30f rep1 − f LO , and 30f rep2 − f LO were simultaneously acquired by a digitizer (resolution = 14 bit, sampling rate = 20 MHz). From the temporal waveforms, we determined instantaneous values of f rep1 , f rep2 , f beat1 , and f beat2 using the instantaneous-frequency-calculation algorithm 8 . Finally, we determined f THz by substituting them into Eqs. (4 to 6). Since the CW-THz test source, the local oscillator, and the clock signals of the digitizer shared a common time-base signal from a 10 MHz rubidium (Rb) frequency standard (Stanford Research Systems FS725, accuracy = 5 × 10 -11 , stability = 2 × 10 −11 at 1 s), one can evaluate the relative precision of frequency measurement without the influence of the absolute precision of the frequency standard.
To confirm the three situations in Fig. 1, we measured f beat1 and f beat2 when f THz was set at (a) 100,013,820,000 Hz for f THz < mf rep1 < mf rep2 , (b) 100,016,340,000 Hz for mf rep1 < f THz < mf rep2 , and (c) 100,020,240,000 Hz for mf rep1 < mf rep2 < f THz . Figure 6 shows the temporal change of f beat1 and f beat2 , where their frequency deviations from the initial values are indicated by δf beat1 and δf beat2 , when (a) f THz < mf rep1 < mf rep2 , (b) mf rep1 < f THz < mf rep2 , and (c) In all graphs, f beat1 and f beat2 fluctuated monotonically due to the drift of f rep1 and f rep2 in the free-running operation. However, the directions of the temporal fluctuations were different from each other. In Fig. 6(a and c), f beat1 and f beat2 indicated similar behaviour to each other, namely, a monotonic decrease or increase. On the other hand, in Fig. 6(b), f beat1 and f beat2 changed in the opposite directions to each other, while their sum remained constant. These behaviours correctly reflect three situations in Fig. 1 and Eq. (4). Finally, we could correctly determine m to be all 3,119 in Fig. 6(a,b and c) based on Eqs (4 to 6).
Next, we measured f rep1 , f rep2 , f beat1 , and f beat2 when f THz was fixed at 100,020,240,000 Hz. After acquiring the temporal waveforms for f rep1 , f rep2 , f beat1 , and f beat2 at a sampling rate of 20 MHz, we calculated their mean values every 10 ms, which corresponds to a measurement rate of 100 Hz. Figure 7(a,b,c and d) show the temporal changes of the mean values for them. All values temporally fluctuated due to the free-running behaviour of the laser rather than the fluctuation of f THz . By substituting f rep1 , f rep2 , f beat1 , and f beat2 in Eqs (4 to 5), the value of m was determined to be 3,119, as shown in Fig. 7(e). Finally, from Eq. (6), we determined the mean and standard deviation of f THz to be 100,020,239,860 Hz and 125 Hz in repetitive measurements of f THz at a measurement rate of 100 Hz, as shown in Fig. 7(f). Therefore, the relative accuracy and precision of the absolute frequency measurement were 1.4 × 10 −9 and 1.2 × 10 −9 , respectively.   Figure 8 shows the measurement precision with respect to the measurement rate and the corresponding measurement time. The measurement precision and the measurement rate showed a trade-off relation within a range of measurement rates from 1 to 100 Hz. However, the correct determination of f THz was impossible at measurement rates higher than 100 Hz, because the measurement error of the numerator | ± f beat2 ± f beat1 | over the denominator f rep2 − f rep1 in Eq. (5) makes it impossible to determine m correctly.
Finally, we performed real-time monitoring of f THz when f THz was changed suddenly or slightly. Figure 9 shows the measured f THz when the nominal frequency of the CW-THz test source was first set at 79,626,000,000 Hz, increased by 20,395,080,000 Hz, decreased by 513,120,000 Hz, and then increased by 2,020,260,000 Hz. The measured f THz at each frequency setting was determined to be 79,626,000,029 ± 47 Hz, 100,021,079,989 ± 32 Hz, 99,507,959,988 ± 26 Hz, and 101,528,219,978 ± 36 Hz, respectively. Even though f THz changed across many modes in the dual PC-THz combs, f THz was determined correctly.

Discussion
One may wonder why such high precision was achieved in the real-time measurement of f THz by using the dual-PC-THz combs without the stabilization of f rep1 and f rep2 . The reason is that each PC-THz comb always functions as a frequency ruler with equal intervals and a linear scale regardless of whether or not f rep1 and f rep2 are stabilized. Such characteristics are inherent in frequency combs. Only if the temporal waveforms for f rep1 , f rep2 , f beat1 , and f beat2 , are acquired synchronously, f THz can be determined without the influence of unstabilized f rep1 and f rep2 , as demonstrated in Figs 7(f) and 9.
The precision of 1.2 × 10 −9 was achieved at a measurement rate of 100 Hz in the present setup; however, it was 100-times worse than that of the previous experiment with two independent free-running mode-locked lasers 10 .
In the instantaneous-frequency-calculation algorithm 8 , the precision is largely influenced by the signal-to-noise ratio (SNR) of the beat signals with f beat1 and f beat2 10 . The beat signals measured by LT-InGaAs/InAlAs PCAs in the present setup showed the much lower SNR than the signals measured by LT-GaAs PCAs in the previous setup due to high dark-current noise in the LT-InGaAs/InAlAs PCAs (not shown). Therefore, the difference in precision between them arises from the low SNR in beat signals rather than use of the free-running dual-λ mode-locked Er:fibre lasers. In other words, there is still some room to enhance the precision by improving the PCAs.  ∆f rep (= 1.63 kHz) in the dual-λ mode-locked fibre laser used here was relatively high compared with that (typically, less than several tens Hz) in dual mode-locked lasers used in the previous research 10 . In this case, we cannot neglect the dead band in the determination of m. In Fig. 1 and Eqs. (1 to 6), it is assumed that the beat signals at the lowest frequency (freq. = f beat1 and f beat2 ) are generated by the same mode number m of dual PC-THz combs (freq. = mf rep1 and mf rep2 ). The dead band is generated when f beat1 and f beat2 are generated by different mode numbers of the dual PC-THz combs. Figure 10(a) shows the optical spectrum when f THz exists within the dead band, namely In this case, f beat1 is generated by photoconductive mixing between f THz and (m + 1)f rep1 , whereas f beat2 is generated by photoconductive mixing between f THz and mf rep2 . The dead bandwidth ∆f dead is given by The simplest way to reduce the dead band is to reduce ∆f rep . There is still some room to further reduce ∆f rep of the dual-λ mode-locked fibre laser down to a few hundred Hz by optimizing the fibre length and dispersion. In this case, it is expected that ∆f dead can be reduced to around 0.5 MHz, which corresponds to 1.6% of the measurement window. Work is in progress to develop a dual-λ mode-locked fibre laser with lower ∆ f rep .

Conclusions
We measured the absolute frequency of CW-THz radiation using dual PC-THz combs induced by a dual-λ mode-locked fibre laser. To the best of our knowledge, this is the first time such a laser system has been employed for frequency measurement in THz region. Although this laser was operating in the free-running mode without stabilization of f rep1 and f rep2 , a relative precision and accuracy of 1.2 × 10 −9 and 1.4 × 10 −9 were achieved at a measurement rate of 100 Hz due to the common-mode behaviour of f rep1 and f rep2 , in addition to the fact that the interval between the PC-THz comb modes was kept equal regardless of the fluctuation in f rep1 and f rep2 . Furthermore, an abrupt or slight change in f THz could be accurately monitored due to the real-time capability thanks to the use of dual PC-THz combs. Although the dual-λ mode-locked fibre laser was used in this work for measuring the frequency of CW-THz radiation in real time, it should be possible to apply it to THz spectroscopy and other metrology applications based on dual THz combs, such as ASOPS THz time-domain spectroscopy [21][22][23][24] , dual THz comb spectroscopy [25][26][27] , and ASOPS THz impulse ranging 28 . In particular, the constant ∆f rep in the free-running operation will enable correct scale conversion of the time axis or frequency axis in these spectroscopic applications. This dual-λ mode-locked fibre laser will open the door to enhance versatility and practicability in dual-THz-comb-based THz measurement systems.

Methods
Free-running, dual-λ mode-locked Er:fibre laser oscillator. As shown in Fig. 2(a), the free-running, dual-λ mode-locked Er:fibre laser oscillator consists of a 980-nm pumped laser diode (LD), a 980/1550 nm wavelength-division multiplexer (WDM), a single-mode fibre (SMF), a 2 meter length of erbium-doped fibre (EDF, Changfei 1022), a polarization-independent optical isolator (ISO), a home-made single-wall carbon nanotube saturable absorber (SWNT-SA), a fibre-squeezer-based polarization controller (PC), a 90/10 fibre output coupler (OC), and an in-line polarizer (ILP) with two 0.25-meter-long polarization maintaining fibre (PMF) pigtails at both ends. The lengths of the commercial single-mode fibres (SMF-28 and HI 1060) in the cavity were estimated to be ~3.4 m and ~0.35 m, respectively, and therefore, the total dispersion was estimated to be ~0.063 ps/nm. The SWNT-SA had a transmittance of 24% at 1540 nm and was fabricated on an FC/APC ferrule from a ~0.27 wt% SWNT solution by using the optical deposition method. By introducing the ILP with its transmission aligned along the slow axis of the PMF into the ring fibre laser, birefringence-induced filtering and loss control effects enabled multi-wavelength lasing in the cavity 12,13,19 . With the adjustment of the intracavity polarization state, simultaneous mode-locking centred on the 1530-nm and 1560-nm regions could be realized. ∆f rep was related to the cavity dispersion of the fibre laser, whereas f rep1 and f rep2 were related to be the fibre length; their values can be further adjusted by optimizing the fibre length and dispersion. Figure 5 shows a schematic diagram of the setup for measuring the frequency of CW-THz radiation The amplified λ 1 pulse light at f rep1 from one EDFA (EDFA1) was collimated in free space and then focused onto a gap in a free-space-coupled, bowtie-shaped, low-temperature-grown (LT) InGaAs/InAlAs PCA (PCA1, TERA15-BT3, Menlo Systems) by a lens (L), whereas the amplified λ 2 pulse light at f rep2 from the other EDFA (EDFA2) was directly fed into a fibre-coupled, dipole-shaped LT-InGaAs/InAlAs PCA detector (PCA2, TERA 15-RX-FC, Menlo Systems) via an optical fibre. This resulted in the generation of dual PC THz combs: PC-THz comb 1 with a frequency spacing f rep1 in PCA1 and PC-THz comb 2 with a frequency spacing f rep2 in PCA2.

Real-time determination of f THz with dual PC-THz combs.
The CW-THz test source was an active frequency multiplier chain (Millitech AMC-10-R0000 with multiplication factor = 6, tuning range = 75-110 GHz, and mean power = 2.5 mW), which amplified the output frequency of a microwave frequency synthesizer (Agilent E8257D, linewidth < 0.1 Hz) by a factor of six. Since this test source was phase-locked to a 10 MHz rubidium (Rb) frequency standard (Stanford Research Systems FS725, accuracy = 5 × 10 −11 , stability = 2 × 10 -11 at 1 s), its output was CW-THz radiation with a linewidth of less than 0.6 Hz and a frequency accuracy similar to that of the frequency standard. When the CW-THz radiation was incident on both PCA1 and PCA2, photoconductive mixing between the CW-THz radiation and the dual PC-THz combs resulted in the output of a current signal from them. The current signals from PCA1 and PCA2 were amplified and filtered by current preamplifiers (AMP, bandwidth = 10 MHz, transimpedance gain = 10 5 V/A), and the beat frequencies below half of f rep1 or f rep2 were extracted as f beat1 and f beat2 .
Portions of light from the EDFAs were detected with photodetectors (PD, Thorlabs DET01CFC, freq. bandwidth = 1.2 GHz). Since the output signal from the PDs included a fundamental component and a series of harmonic components of f rep1 or f rep2 within the frequency bandwidth of the PDs, we selected the 30-th harmonic component of f rep1 or f rep2 , namely 30f rep1 and 30f rep2 , in order to magnify the frequency fluctuation. The components 30f rep1 and 30f rep2 were electrically mixed with an output signal from a local oscillator (LO, f LO = 961,000,000.00 Hz) using a double-balanced mixer (M), and the resulting beat signals 30f rep1 − f LO and 30f rep2 − f LO were extracted by two low-pass filters (LPF). Temporal waveforms for f beat1 , f beat2 , 30f rep1 − f LO , and 30f rep2 − f LO were simultaneously acquired by a digitizer (resolution = 14 bit, sampling rate = 20 MHz). From the temporal waveforms, we determined instantaneous values of f rep1 , f rep2 , f beat1 , and f beat2 using the instantaneous-frequency-calculation algorithm involving a Fourier transform, digital frequency filtering, an inverse Fourier transform, a Hilbert transform, the time differential of the instantaneous phase, and signal averaging 8 . Finally, we determined f THz by substituting these values into Eqs. (4 to 6). Since the CW-THz test source, the local oscillator, and the clock signals of the digitizer shared a common time-base signal from the frequency standard, one can evaluate the relative precision of frequency measurement without the influence of the absolute precision of the frequency standard.