The terahertz electromagnetic wave is located between millimeter wave and infrared light, with the frequency range roughly defined from 0.1 to 10 THz^{[1, 2]}. Given its unique spectral characteristics, the terahertz wave holds significant potential for a wide range of applications such as imaging, environmental monitoring, medical diagnosis, and communications^[3].

Frequency comb^{[4, 5]} refers to a coherent light source that emits a broad spectrum of equally spaced frequency lines. The frequency of each comb line (f_N) can be precisely determined by two frequencies, i.e., f_N = fceo + N·f_rep, with f_ceo and f_rep being carrier offset and repetition frequencies, respectively. Therefore, by controlling f_ceo and f_rep, we can firmly stabilize a frequency comb. Frequency combs have been widely utilized in various fields, such as metrology, spectroscopy, astronomy, atomic clocks, precision ranging, and so on, owing to their stable frequency lines and ultrashort optical pulses^[6−9]. The direct application of frequency combs is the dual-comb spectroscopy which employs two frequency combs with slightly different repetition frequencies. Under dual-comb operation, the beating between two frequency combs in the optical range can produce a new comb in the microwave band^[10]. The dual-comb spectroscopy can significantly enhance the speed and accuracy of spectroscopic measurements^[11].

In the terahertz frequency range, the electrically pumped quantum cascade laser (QCL) offers several advantages, e.g., small size, high power, and low beam divergence^[12−14], making it an ideal radiation source for generating frequency combs emitting between 1 and 5 THz^[15−17]. In free-running terahertz QCLs, frequency combs can be generated by exploiting the nonlinear four-wave mixing effect of the laser gain medium, combined with the optimal design of group velocity dispersion (GVD) and dispersion compensations^{[18, 19]}. However, the large phase noise of free-running terahertz QCLs significantly affects the precision of frequency combs in various applications^{[20, 21]}. Hence, evaluating the phase noise of terahertz QCL frequency comb and dual-comb sources holds great importance for high-precision measurements.

In this work, we present a detailed analysis of the phase noise characteristics of terahertz QCL frequency comb and dual-comb sources emitting around 4.2 THz. We first measure the current noise of different models of direct current (DC) sources and the phase noise of different radio frequency (RF) sources. The current noise generated by the DC sources is extremely low, at the level of nA/Hz^1/2. Such a level of current noise can be safely disregarded for evaluating the noise characteristics of frequency comb and dual-comb signals. Then, we select one dual-comb line for phase locking and measure its phase noise with and without the phase-locked loop (PLL). The results reveal that a significant reduction in phase noise in the low offset frequency band can be achieved by employing the phase locking technique. By comparing the phase noise of the RF sources with those of the intermode beatnote and dual-comb signals, it is indicated that the phase noise of the RF sources can be neglected for evaluating the phase noise of the optical system.

Intensity noise and phase noise are two important categories of noise in a laser system. Intensity noise refers to the random variations in the amplitude of a laser, whereas phase noise pertains to the random fluctuations in the phase. To analyze the frequency comb noise, the primary focus lies in examining phase jitter, which manifests as time jitter in the time domain and frequency jitter in the frequency domain. Since each frequency line in a comb laser is determined by two frequencies, i.e., f_ceo and f_rep, the phase noise is also resulted from two noise components, i.e., carrier offset and repetition phase noise. Here, we pay more attention on the phase noise of f_rep. Phase noise is usually measured as the power spectral density (PSD) of the phase fluctuations, usually expressed in units of dBc/Hz (decibels relative to the carrier per hertz) at a given offset from the carrier frequency. Unlike spectral linewidth, the phase noise PSD provides more detailed information. According to the semiclassical theory of lasers, the laser field can be represented as:

1E(t)=(E0+a(t))ei(w0t+∅(t)),

where a(t) and ∅(t) denote the amplitude and phase jitters, respectively. The autocorrelation function of the optical field RE(τ) is given by^[22]:

2RE(τ)=〈E(t)E*(t+τ)〉=limT→∞1T∫−T2T2E(t)E*(t+τ)dt=〈eiΔφ(t,τ)〉eiw0τ,

where T is the measurement time, 〈Δφ(t,τ)〉 denotes the phase jitter of the light field during the period from t to t+τ. Applying the Wiener−Khinchine theorem, we can perform a Fourier transform on the autocorrelation function of the phase variable Δφ(t,τ) to obtain the phase noise PSD S∅(f):

3S∅(f)=∫−∞∞e−i2πfτ〈(Δφ(t,τ))2〉dτ,

with f being the frequency.

The phase noise observed in terahertz QCL frequency combs can be attributed to various factors, including environmental noise such as temperature fluctuations and mechanical vibrations, as well as pump source intensity noise and noise resulting from spontaneous radiation effects. The impact of temperature variations, mechanical vibrations, and other environmental noises is prominent in the low-frequency range. This type of noise is commonly referred to as 1/f noise or flicker noise, and its magnitude decreases as the offset frequency increases. Spontaneous radiation in a semiconductor laser generates white noise, also known as quantum noise, which primarily occurs in the high-frequency range. The phase PSD of white noise remains constant and independent on frequency. It is considered to be intrinsic noise associated with the semiconductor laser and serves as a fundamental factor in determining the limit of the laser linewidth. The phase noise gradually converges to the level of spontaneous radiation noise as the frequency increases. Alongside the aforementioned noise sources, several other factors can impact the phase noise of frequency combs. These factors include the laser drive sources, RF generators, laser gain processes, detector characteristics, transmission environment, etc. Although these extrinsic factors can contribute to an increased noise floor and reduced signal-to-noise ratio, their influence is generally less significant compared to noise sources like temperature fluctuations, mechanical vibrations, and spontaneous radiation effects. It is crucial to consider and mitigate these various noise sources to minimize their impact on the phase noise of frequency combs.

Fig. 1(a) shows the experimental setup employed for phase noise measurements of terahertz QCL frequency comb and dual-comb signals. The two terahertz QCLs utilized in this experiment are based on a hybrid active region design that combines the optical transition of the bound-to-continuum and resonant phonon scattering for depopulation in the lower laser state^[23]. The entire active region of the QCL was grown on a semi-insulating GaAs (100) substrate using molecular beam epitaxy. The grown wafer was then processed into a single-plasmon waveguide structure. It is worth nothing that in this work the ridge width and cavity length for both terahertz QCLs are 150 μm and 6 mm, respectively, since these two dimensions are more favorable for frequency comb operation^{[19, 22]}. For the dual-comb operation, the two terahertz QCL combs are positioned face to face on a Y-shaped sample holder, with a separation distance of 34 mm between their emitting facets. The multiheterodyne dual-comb signals are measured using one of the terahertz QCL combs as a fast terahertz detector (self-detection scheme) without a need of an external fast terahertz detector^[24]. This particular geometry configuration eliminates the need for optical alignments, ensuring that both terahertz QCLs experience identical physical environments throughout the entire measurements^[25]. Two low-noise current sources, DC-1 and DC-2, are used to provide continuous electrical pumps to the terahertz QCLs for lasing. The amount of current supplied to the terahertz QCLs determines their operating states, either in single-mode or frequency comb operation mode. In this work, both terahertz QCLs are operated as frequency combs. As shown in Fig. 1(a), both terahertz QCLs, Comb1 and Comb2, emit terahertz photons. The multiheterodyne beating (dual-comb) signals between the two combs are detected by Comb1. The photocurrent of Comb1 including the intermode beatnote and dual-comb components is sent into a bias-T and then amplified by 30 dB before being displayed on a spectrum analyzer (R&S, FSW26). The phase noise spectra of these signals are measured using a phase noise module (B61) of the phase noise analyzer (R&S, FSWP 26).

Figs. 1(b) and 1(c) show the intermode beatnote maps for Comb1 and Comb2, respectively. We can see that as the current exceeds 900 mA for Comb1 and 850 mA for Comb2, both lasers operate in the frequency comb state demonstrating single narrow intermode beatnotes. To obtain stable dual-comb operation, we operate the two terahertz QCLs at 1000 and 920 mA as marked in Figs. 1(b) and 1(c), respectively. In Figs. 1(d) and 1(e), we show the measured intermode beatnote and dual-comb spectra when the two terahertz QCLs are operated at 1000 and 920 mA, respectively. The intermode beatnote signals (repetition frequencies) of Comb1 and Comb2 are denoted as f_rep1 and f_rep2, respectively. Due to slight variations in material growth, fabrication processes, drive currents, etc., the repetition frequencies of the two terahertz frequency combs are not exactly identical^[16]. In this case, there is a difference of 16.74 MHz between the repetition frequencies. Note that this small difference in repetition frequencies is critical for dual-comb operation. In a dual-comb spectrum, the line spacing is equal to the frequency difference between the two repetition frequencies. Therefore, if the two repetition frequencies are the same, no dual-comb signals can be observed. As shown in Fig. 1(e), the dual-comb spectrum shows a line spacing of 16.42 MHz which is nearly equal to the difference in repetition frequency shown in Fig. 1(d). The small discrepancy of ~300 kHz is because the two spectra in Figs. 1(d) and 1(e) were not measured at the same time and the two laser combs were operated in free-running mode demonstrating a natural frequency instability (drift) of hundreds of kilohertz.

To overcome the above-mentioned frequency instability issue, in this work we also implement the phase locking of the intermode beatnote and dual-comb signals by employing a PLL. Phase locking of the intermode beatnote signal is relatively straightforward. By beating the terahertz QCL with the referenced RF signal, and then feeding the beatnote signal into a PLL. The detailed method can be found in Ref. [26]. For the phase locking of the dual-comb signal, as shown in Fig. 1(a), the selected dual-comb line is down-converted, filtered, and amplified before being sent to the PLL for phase locking. The error signal from the PLL is conveyed to the DC side of the terahertz QCL to dynamically control the drive current, thereby locking the frequency of the dual-comb line to the frequency of the local oscillator (LO)^{[16, 23]}. By adopting this approach, the fluctuation of the signal is suppressed and its phase noise is reduced.

Before examining the phase noise properties of the intermode beatnote and dual-comb signals, we first measure the noise spectra of different DC and RF sources used in the experiment. As shown in Fig. 1(a), DC-1 (Keysight E3644A) and DC-2 (QubeCL) are low-noise DC sources employed to drive terahertz QCLs. The noise present in the output current is a combination of thermal noise in the resistor, shot noise in the semiconductor, flicker noise (1/f), electromagnetic interference caused by external electromagnetic fields, and switching noise caused by the switching action of the switching elements. Excessive current noise can lead to a degradation in overall system performance. It is crucial to understand the noise level of the DC sources used in the system to ensure accuracy and reliability of the system.

For the current noise measurement of the DC sources shown in Fig. 1(a), the output of the current sources is directly connected to the baseband input of the phase noise analyzer. The measurement results obtained on the phase noise analyzer are typically expressed in dBm/Hz. By utilizing the relationships between dBm and power in watts (W), as well as the relationship between power and current, it is straightforward to convert the measurement results from dBm/Hz to nA/Hz^1/2. The converted current noise power spectral density is shown in Fig. 2. We compare the current noise spectra of two current sources, i.e., DC-1 (Keysight E3644A) and DC-2 (QubeCL), operated at different drive currents. The measurement results in Fig. 2 indicate that DC-2 demonstrates a lower current noise compared to DC-1. For example, at 100 Hz and 1 kHz, the measured current noise for DC-1 is 19.16 and 5.17 nA/Hz^1/2, respectively, while they are measured to be 0.27 nA/Hz^1/2 at 100 Hz and 0.09 nA/Hz^1/2 at 1 kHz for DC-2. It should be noted that both DC sources exhibit current noise levels at the nA/Hz^1/2 range, demonstrating excellent low-noise performance. In terahertz QCL frequency comb and dual-comb systems, the current noise generated by the DC sources can be safely neglected.

In addition to the DC source, the RF sources used in the experiment can also introduce noises. The electronic devices and components in the RF sources contribute thermal noise, 1/f noise, and quantum noise. These noises cause phase instability in the oscillator output, resulting in phase noise. The phase noise PSD of the RF sources can be directly measured using a phase noise analyzer. Fig. 3 shows the measured phase noise spectra of three RF sources with different output power values. It can be seen that for each RF source, the low power RF signal has worse phase noise, especially when the offset frequency is greater than 100 Hz. In the power range investigated here, the 0 dBm signal shows the best phase noise behavior for all three RF sources. Table 1 summarizes the specific phase noise values at different offset frequencies for the three RF sources at 0 dBm. It is evident that different RF sources exhibit distinct phase noise performances. Specifically, RF-1 (R&S SMA 100B) generates lower phase noise compared to the other two RF sources. This observation emphasizes the crucial role of comprehending the noise characteristics of experimental equipment. With a comprehensive understanding of these characteristics, we can select equipment with lower phase noise, thereby minimizing the impact of external interference.

In Fig. 4, we show the experimental results of phase locking of the terahertz dual-comb system. Fig. 4(a) presents the terahertz dual-comb spectrum around 4.6 GHz, with one of the dual-comb lines at 4.58 GHz (indicated by the red arrow) being phase locked. A stable RF-1 reference signal at 4.4898 GHz is the injecting RF signal employed to down-convert the dual-comb signal to 95 MHz. A bandpass filter is used to select one line from the down-converted dual-comb lines, which is then sent to the PLL for phase locking. The spectrum of the phase locked line at 93.62 MHz is shown in Fig. 4(b). The inset of Fig. 4(b) is the high-resolution spectrum of the phase locked line measured with a RBW of 1 Hz and a VBW of 1 Hz. We can clearly see that the measured signal-to-noise ratio (SNR) is 51 dB, indicating a good locking of the dual-comb line.

Note that the intermode beatnote signal contains both the phase and noise information of an individual frequency comb, making it a valuable tool for assessing and characterizing the stability of the frequency comb. To gain a comprehensive understanding of the stability of the terahertz QCL frequency comb and dual-comb system, we measure the phase noise spectra of the free-running intermode beatnote and dual-comb signals. The results are shown in Fig. 5. It can be seen that in free-running mode the measured phase noise of the intermode beatnote signals (f_rep1 or f_rep2) is always lower than that of the dual-comb line (marked by the red arrow in Fig. 4(a)) in the entire frequency range. This can be understood by the generation process of the dual-comb signal which involves multiheterodyne mixing of two frequency combs. The noise accumulation of two individual combs and the down-conversion process which introduces additional noise, finally result in larger phase noise for the free-running dual-comb signal. It is worth noting that for the two intermode beatnote signals, f_rep1 and f_rep2, they demonstrate different phase noise levels in free-running mode, i.e., f_rep1 exhibits higher phase noise compared to that of f_rep2, although the two lasers are designed to be nominally identical. It can be found that for the three signals in free-running, their phase noise shows similar dependence as frequency: initially, the phase noise gradually decreases with increasing frequency; in the frequency range between 30 Hz and 30 kHz, the phase noise decreases rapidly with frequency, following a 1/f ⁵ dependence; as the frequency is greater than 30 kHz, the phase noise exhibits a 1/f² dependence that is primarily induced by external environmental factors, such as temperature fluctuations, current fluctuations, and mechanical vibrations; when the frequency exceeds 2 MHz, the phase noise tends to flatten out and is primarily dominated by white noise. At 2.5 MHz, the measured phase noise is approximately −102 dBc/Hz. As a reference, phase noise spectra of the RF sources (RF-1, RF-2) used in the experiment are also displayed in Fig. 5. At a frequency of 10 kHz, the phase noise of the intermode beatnote (f_rep2) is −70 dBc/Hz, while the phase noise of the two RF sources is around −120 dBc/Hz. Therefore, the phase noise generated by the RF sources can be completely negligible.

To quantify the locking effect, in Fig. 5 we also show the phase noise spectra of the locked intermode beatnote (f_rep2) and dual-comb line at 4.58 GHz. As the offset frequency is smaller than 45 kHz, the phase locking demonstrates significant function to suppress phase noise of the intermode beatnote and dual-comb signals. At 100 Hz, the measured phase noise values of the dual-comb line without and with PLL are 15.026 and −64.801 dBc/Hz, respectively. In particular, in the frequency range between 1 and 10 Hz, the phase noise of the locked dual-comb line reaches the level of the reference RF source. It indicates that at the small offset frequencies when the PLL is switched on the high stability of the RF source can be transferred to the locked dual-comb signals. In Fig. 5, we also show the phase noise spectra of the locked intermode beatnote signals (f_rep2). It can be observed that the suppression effect of phase locking on the intermode beatnote signal is in line with that of the dual-comb line. Therefore, we can conclude that phase locking has a notable impact in reducing the phase noise.

In conclusion, we characterized the phase noise of terahertz QCL frequency comb and dual-comb sources operating around 4.2 THz, and analyzed the sources of phase noise in different Fourier frequency regions. Understanding the phase noise behavior of frequency comb and dual-comb sources is crucial for evaluating their performance and limitations. By measuring the current noise of DC sources and the phase noise of RF sources, we can exclude their influences on the phase noise of the frequency comb and dual-comb sources. Additionally, we investigated the impact of phase locking on the phase noise of terahertz dual-comb. The results demonstrated that the phase locking technique could substantially suppress the phase noise in the low-frequency band. Overall, the paper provides detailed information on the phase noise characteristics of terahertz QCL frequency comb and dual-comb. This serves as a valuable reference for future studies on phase noise in terahertz QCL frequency comb and dual-comb and their applications.