1. INTRODUCTION

The locking of one oscillator to another when there is a degree of coupling between them is a universal phenomenon that was first observed by Huygens in 1665 and later described by Adler [1]; it applies equally well to mechanical, electronic, and optical oscillators. For most applications, a weaker “master” oscillator with desirable temporal and spectral characteristics is used to control the output of a more powerful “slave” oscillator by coupling, or “injecting,” the “master” oscillation into the “slave” oscillator. This technique has become ubiquitous in many areas of science and technology, and particularly in (ultra-)high-resolution spectroscopy and for the distribution of frequency standards.

High-resolution spectroscopy is a fundamental technique that finds application across a wide range of science and technology. It can be used to identify atoms and molecules through spectral fingerprinting; provide richly detailed information on the immediate chemical and physical environment, and relative motion, of the chemical species; and test fundamental theories of matter. Although high-resolution spectroscopy is well established in the visible and near-infrared (near-IR) regions of the spectrum, such application in the far-IR or terahertz (THz) region is poorly developed because of the lack of compact, stable, narrowband sources. To give one important example, astronomical spectroscopy commonly uses heterodyne detection to measure THz frequency emission from distant objects [2,3]. This passive technique uses a mixer and a local oscillator (LO) to measure weak signals close to the frequency of the narrow linewidth LO. Traditionally, LO THz sources based on gas lasers have been used, but these are bulky, confined to particular gas lines, and difficult to package in a form suitable for satellite-based instruments.

Another area of contemporary importance that requires high-resolution THz frequency sources is in the trapping and cooling of molecules [4,5]. Ultra-high-resolution spectroscopy of ultra-cold molecules in the THz spectral region would enable the investigation of fundamental physical constants [6,7], the particle physics standard model [8], and help determine molecular structure [9].

For these reasons, there has been significant interest in improving both the resolution and compactness of spectrometry in the THz spectral region. For frequencies below $\sim 1.5\mathrm{THz}$, frequency multiplier chains have been used both as LOs in heterodyne spectroscopy [10] and as narrowly tunable sources for transmission spectroscopy of cold molecules [11,12]. There has also been use of continuous-wave (CW) THz photomixers to provide emission at 0.6 THz and 1.2 THz, driven by a pair of diode lasers [13]; here the driving lasers were stabilized using a wavelength meter to achieve a THz emission linewidth of 4 MHz.

In the spectral region from 1.5 to 5 THz, however, the power available from frequency multiplier chains becomes impracticably small, and, as such, the most promising source is the THz quantum cascade laser (QCL) [14]. QCLs are electrically pumped semiconductor lasers operating across the mid-IR and THz spectral regions. They are compact, are high power [15], can be engineered to operate broadband or single mode, and can also show spontaneous frequency comb operation [16]. The linewidth of a QCL is intrinsically narrow [17] and exhibits a near-zero linewidth enhancement factor [18,19], but a free-running device has typically a much broader linewidth, around 1–2 MHz over short time scales [20] and up to 10 MHz over several seconds [21]. It therefore requires stabilization or locking for high-resolution applications.

To date, locking of THz QCLs to a gas line [22], a frequency multiplier chain [23,24], a femtosecond laser [20], and a femtosecond laser comb [25] has been demonstrated, and very recently, the locking of a mid-IR QCL to a frequency comb has been shown, with the latter derived from a traceable frequency standard and based on a nonlinear difference frequency generation technique [26]. However, all of these demonstrations made use of a negative feedback loop with a limited loop-bandwidth (a phase-locked loop) to provide a negative feedback voltage onto the QCL rather than the direct injection locking observed by Huygens. A direct injection approach has a simpler implementation, does not require a sensitive detector, overcomes the requirement for low loop delay, and can compensate for phase fluctuations over a wide bandwidth. Furthermore, the need to lock to a gas line or a femtosecond laser via a phase-locked loop results in a system that is either bulky, limited to particular frequencies, or both, which constrains its use for many potential applications.

In this work, we demonstrate injection locking of a QCL operating at a frequency around 2 THz to an all-fiber telecommunications-frequency comb, which has great potential for compact integration in photonic circuits. Locking is achieved by selecting a pair of comb lines from the near-IR comb and generating a difference frequency injection locking signal from them, close to the frequency of the free-running QCL. This THz difference frequency is then coupled into the QCL to injection lock it.

2. RESULTS

Our near-IR comb source uses a narrow linewidth C-band laser to seed an amplified fiber loop incorporating a high-speed phase modulator, driven by a microwave frequency synthesizer that determines the comb line spacing (see Supplement 1) [27]. The fiber loop operates as a resonant cavity and multiplies the number of comb lines produced by the phase modulator. The resulting all-fiber frequency comb generates over 125 lines and spans $>2.7\mathrm{THz}$, centered around 1552.5 nm. The comb line spacing is exactly equal to the microwave modulation frequency, $f_{\mathrm{RF}}$, which can be adjusted in discrete steps from 18.5 to 20 GHz (see Supplement 1 for further details). It is important to note that while the linewidth of each comb line is determined by the linewidth of the seed laser (here 15 kHz), for this work, we use the heterodyne between two selected comb lines. For an ideal system, the linewidth of this heterodyne signal is $N$ times the linewidth of the microwave reference ($<1\mathrm{Hz}$), where $N$ is the number of comb lines separating the two selected lines.

The experimental arrangement used for injection locking the THz QCL is shown in Fig. 1. The microwave frequency synthesizer is set to $f_{\mathrm{RF}}=19.773076000\mathrm{GHz}$ to control the spacing of the near-IR comb. The comb signal is split and filtered to obtain two comb lines that are separated by a frequency $f_{\mathrm{THz}}$ close to the THz QCL frequency ($\sim 1.97\mathrm{THz}$). For $N=101$, we find $f_{\mathrm{THz}}=1997.080676\mathrm{GHz}(f_{\mathrm{THz}}=101\times f_{\mathrm{RF}})$, and this is the THz reference frequency to which the QCL is locked. The two comb lines required are selected by a wavelength selective switch, and each selected line is used to injection lock a separate tunable telecommunications wavelength laser (DS-DBR laser from Oclaro Technologies, UK). The signals from these lasers are then combined, amplified by an erbium-doped fiber amplifier (EDFA), and split by a 50:50 optical power splitter, with one portion sent to a fiber-coupled CW THz transmitter (Tx) and the second portion sent to a fiber-coupled CW THz receiver (Rx) (both supplied by Toptica Photonics; further details are in Supplement 1). A variable delay can be introduced into the receiver arm to control the relative phase between the THz signal arriving at the receiver from the QCL and the near-IR signal beating at THz reference frequency. The current induced in the receiver can be expressed as

 

Fig. 1. Experimental arrangement used for injection locking the THz QCL. Electronic connections are shown in black, fiber connections and near-IR are in red, and free-space THz radiation is in green. Also shown, from the top left, are the RF signal generator (RF Gen); the wideband comb source (COMB); tunable bandpass filters; tunable DBR lasers, laser 1 and laser 2; polarization controllers (PC); an erbium-doped fiber amplifier (EDFA); the continuous-wave THz transmitter (Tx); a function generator (FG); the quantum cascade laser (QCL); a free-space optical delay line for phase adjustment ($\mathrm{\Delta }\psi$); the continuous-wave THz receiver (Rx); a transimpedance amplifier (TIA); and an oscilloscope (SCOPE) for data acquisition.

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The QCL chosen for this work was a bound-to-continuum design that operates around 1.97 THz [31]. The QCL was fabricated with a single plasmon waveguide [32] and mounted on the cold finger of a continuous-flow helium cryostat, which was maintained at a temperature of 14 K. The electrical and optical characteristics of the device are shown in Fig. 2. To tune the frequency of the QCL to be close to $f_{\mathrm{THz}}$, the operating current was chosen to be close to 775 mA, where the emission, measured by a high-resolution Fourier transform infrared spectrometer (FTIR), was single mode at 1.997 THz, marked by a blue dashed line in the figure. Further details on the QCL device are given in Supplement 1.

 

Fig. 2. I–V characteristics and THz output of the QCL. The device is operated in continuous wave with a heat sink temperature of 14 K. The blue dot marks the operating current used in this work (775 mA). The spectrum acquired at this operating point, when the QCL is free-running, is shown in the inset. The resolution for this spectrum is 250 MHz and shows a single lasing mode.

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After basic characterization, the QCL device was mounted in the experimental arrangement shown in Fig. 1 and carefully aligned to maximize coupling of the CW THz radiation from the transmitter into the QCL cavity. To reduce heating in the transmitter, its bias was pulsed at a frequency of 1.15 kHz, with 50% duty cycle. This also allowed monitoring of the locked and unlocked signals in real time and also provided a reference signal for lock-in detection.

The measured heterodyne beat frequency between the QCL and the THz reference frequency, captured over 10 ms, is shown in Fig. 3 as a function of QCL drive current and, hence, frequency. The heterodyne signal with the THz transmitter turned off is shown in black; the QCL current is tuned to adjust the beat frequency, with a measured tuning sensitivity of 6.5 MHz/mA. The FWHM (3 dB width) of the beat signal is measured to be 1.0 MHz with a 10 dB width of around 1.8 MHz, thought to be limited by noise in the QCL current source (Keithley 2400). The effective linewidth of the reference signal was calculated from the measured linewidth of the microwave frequency synthesizer ($<1\mathrm{Hz}$, limited by the resolution of the measurement) to be $<100\mathrm{Hz}$; therefore, the observed broadening of heterodyne signal can be assumed to arise entirely from the free-running QCL. This is in good agreement with previous measurements of free-running THz QCL linewidths [20], which are much larger than the quantum-limited intrinsic linewidths that THz QCLs of this type have shown [17]. Over longer time scales ($\sim \text{minutes}$), there is a larger drift in QCL frequency caused by thermal fluctuations and mechanical vibrations; these could be reduced by using a bath cryostat rather than the continuous-flow cryostat that we used here.

 

Fig. 3. Beat signal between the reference signal derived from the comb source and the QCL for the THz transmitter is off (black) and on (red). The QCL current is adjusted to tune the QCL frequency. The QCL current is adjusted over a range of 1.5 mA to tune the QCL emission frequency over the 9 MHz range shown in the figure.

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The same axes also show the equivalent spectra when the transmitter is switched on (red). Upon switching the THz transmitter on, a significant amount of frequency pulling is observed immediately, with a frequency jump of $\sim 2\mathrm{MHz}$ for a frequency difference between the free-running QCL and reference frequency of 10 MHz [Fig 3(a), panel (i)]. As the heterodyne frequency between the QCL and THz reference frequency is reduced by tuning the QCL current, this frequency pulling effect increases until, for frequency differences less than $\sim 5\mathrm{MHz}$ [Fig. 3(a), panels (iv)–(vi)], the heterodyne beat is no longer visible when the transmitter is switched on. At this point, all of the signal appears at zero frequency (DC), and the QCL is injection locked to the reference frequency. The vertical axis of Fig. 3 shows relative power, in dB, to the peak signal at zero frequency when the QCL is locked. The QCL typically remains injection locked for up to few minutes before the lock is lost, thought to be caused by thermal fluctuations outside the locking range.

To obtain an indication of the QCL locking range, $\mathrm{\Delta }{\upsilon }_{\mathrm{lock}}$, we plot the ratio of the integrated noise (0–10 MHz) obtained with the transmitter on and off, as a function of heterodyne peak position for the free-running QCL, Fig. 4(a). The figure shows a marked reduction in noise when the peak of the heterodyne signal is less than 5 MHz, which implies a full width locking range of 10 MHz. Figure 4(b) shows the locking range observed from the QCL for different Tx powers, $P_{\mathrm{Tx}}$. The fit confirms that the expected relationship is observed, $\mathrm{\Delta }{\upsilon }_{\mathrm{lock}}\propto \sqrt{P_{\mathrm{Tx}}}$.

 

Fig. 4. (a) Integrated noise (0–10 MHz) as a function of the heterodyne beat signal position. The shaded region indicates the heterodyne frequencies for where the QCL becomes locked. (b) The observed locking range as a function of Tx power, showing a least-squares fitting to Adler’s equation.

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We can compare the measured locking range of the THz QCL to the expected locking range using the simple expression derived by Adler [1]:

In addition to measuring the heterodyne beat for the locked and free-running QCL, we can make use of the control that is available over the phase of the reference signal to measure both the amplitude and phase of the locked QCL signal (Fig. 5). For this measurement, the length of the optical path to the THz receiver is varied using a free-space optical delay line (labeled $\mathrm{\Delta }\psi$ in Fig. 1). This changes the relative phase of the reference signal with respect to the QCL signal, and by recording this sinusoidal signal for different QCL tunings, the amplitude and phase of the locked QCL signal can be measured. For these measurements, the gain of the transimpedance amplifier was increased to ${10}^6\mathrm{\Omega }$ (bandwidth 1.8 MHz), and the signal was measured using a lock-in amplifier. The lock-in amplifier, together with the reduced transimpedance amplifier bandwidth, ensures that only QCL signals that are locked can be measured. The results are plotted in Fig. 5 for two different injection powers: when the QCL is well locked ($P_{\mathrm{Tx}}=81\mathrm{nW}$) and when only partially locked ($P_{\mathrm{Tx}}=24\mathrm{nW}$). The QCL is partially locked when the locking range is roughly equal to the linewidth such that it is locked and unlocked rapidly. The locked signal amplitude increases as the frequency of the QCL approaches the edge of the locking range. When in the locked range, the amplitude of the signal is flat, within the noise level, but the phase of the signal changes from $-\pi /2$ to $+\pi /2$, as expected from the analysis of Adler [1]. The amplitude of the locked signal ($\sim 90\mathrm{mV}$) agrees with expected value given that the amplitude for the signal from the transmitter is 0.65 mV in this case, giving a power ratio of $\sim \mathrm{19,000}$, close to the expected $P_{\mathrm{QCL}}/P_{\mathrm{Tx}}\sim \mathrm{21,000}$. When the injection power is reduced so that the QCL is never fully locked, the phase change is still observed, but the amplitude does not reach a plateau. The reduced locking range observed in Fig. 5 compared with Fig. 3 is due to the non-optimized alignment of the transmitter with the QCL in the data shown in Fig. 5.

 

Fig. 5. Amplitude and phase of the locked signal measured using a lock-in amplifier. The frequency of the QCL is adjusted via its drive current and converted to frequency using the measured dependence of $-6.5\mathrm{MHz}/\mathrm{mA}$. For each current (frequency) value, the detection phase is scanned and the sinusoidal waveform from the receiver recorded on the lock-in amplifier (shown in the insets). The amplitude and phase of this signal are shown for two injection powers. Because the detection is coherent, only QCL signals phase-locked to the comb will lead to a measurable signal.

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The oscillator noise of the locked QCL shown in Fig. 6 is calculated from the receiver signal when the QCL is locked. For this, the detection bandwidth is increased to 14 MHz here, so both coherent (locked) signals and incoherent (unlocked) signals are measured. Since the noise is calculated from the measured spectrum, it includes components of both amplitude and phase noise. The measured noise when the heterodyne signal lies outside the detection bandwidth ($>20\mathrm{MHz}$) is also shown, to indicate the noise floor of the measurement system, together with the corresponding noise for the unlocked QCL. Apart from a peak at 75 kHz (which is thought to be from the delay stage motion controller) and some further peaks in the 100 kHz–1 MHz range, the phase noise of the locked QCL follows the noise floor of the measurement system over the five decades of frequency shown.

 

Fig. 6. Measured noise for the locked and unlocked QCL. For the locked QCL, two traces are shown, acquired using two different gain (bandwidth) settings for the transimpedance amplifier to cover both low and high frequencies. The noise floor is also indicated, measured by tuning the QCL frequency away from the reference frequency so that the heterodyne beat lies outside the detection bandwidth. The inset shows the heterodyne beat between the locked QCL and the THz reference frequency.

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Finally, to indicate the linewidth achievable from the QCL, we present the heterodyne beat signal between the THz reference frequency derived from the optical comb and the locked QCL in the inset to Fig. 6. The 10 dB linewidth of the beat signal is $\sim 2\mathrm{Hz}$, limited by the resolution of the measurement, implying a sub-Hz FWHM. However, if a Lorentzian lineshape is assumed, and the FWHM is calculated from the phase noise at 100 Hz offset ($\sim -70\mathrm{dBc}/\mathrm{Hz}$), then a value 1 order of magnitude below this ($<100\mathrm{mHz}$) is found. While the absolute linewidth of the locked QCL cannot be reduced below that of the THz reference signal (measured to be $<100\mathrm{Hz}$ in the present case), this indicates the locked QCL linewidths that ought to be achievable with a lower phase noise microwave reference source.

3. CONCLUSION

In conclusion, we have injection locked a THz QCL operating at 2 THz to a near-IR fiber-based optical frequency comb. This not only stabilizes the frequency of the QCL so it becomes traceable to the microwave reference frequency but also allows the phase-locked CW QCL emission to be detected coherently. This brings the same high levels of frequency precision and accuracy that are available in the microwave region to the THz region of the spectrum.

While in this work the comb spacing (and therefore the THz reference frequency) is only tunable in discrete steps, continuous tunability could be implemented with the use of an optical phase-locked loop [34] replacing one of the locked DBR lasers. This would allow continuous tuning of the locked QCL across its full gain-bandwidth. By implementing a slow-feedback to the QCL bias, forming an optical injection phase-locked loop [35], the QCL could be made to track the reference frequency across a large bandwidth. This would also improve long-term stability.

For applications, this scheme is very appealing, particularly since very compact integration should be possible given that all components are semiconductor based; the fiber loop and tunable telecommunications wavelength lasers can be monolithically integrated on-chip as photonic integrated circuits [36]. Furthermore, the QCL and photomixer could be integrated, eliminating the free-space path, making use of only a fiber connection and so reducing the need for alignment.

The work described here enables referencing of THz sources to primary frequency standards in an experimentally convenient format. To use this template, you will need to apply the embedded styles to each paragraph-level item in your manuscript or simply use this template as a visual guide.

The data associated with this paper are openly available from the University of Leeds data repository [37].