Harmonic and Subharmonic RF Injection Locking of THz Metasurface Quantum-Cascade VECSEL

photonics1
Nov. 3, 2024

Abstract

Harmonic and subharmonic RF injection locking is demonstrated in a terahertz (THz) quantum-cascade vertical-external-cavity surface-emitting laser (QC-VECSEL). By tuning the RF injection frequency around integer multiples and submultiples of the cavity round-trip frequency, different harmonic and subharmonic orders can be excited in the same device. Modulation-dependent behavior of the device has been studied with recorded lasing spectral broadening and locking bandwidths in each case. In particular, harmonic injection locking results in the observation of harmonic spectra with bandwidths over 200 GHz. A semiclassical Maxwell-density matrix formalism has been applied to interpret QC-VECSEL dynamics, which aligns well with experimental observations.

Introduction

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Quantum cascade (QC) lasers have been demonstrated as compact, electrically pumped semiconductor frequency comb sources in mid-IR and THz frequency ranges. (1−3) Spontaneous frequency combs have been demonstrated in waveguide-based Fabry-Pérot (4,5) and ring QC-lasers, (6−8) thanks to the inherent strong optical nonlinearity of the gain material; they have been developed into a competitive technology for mid-IR and THz high-resolution, high-speed spectroscopy. Furthermore, different from conventional dense frequency combs, which are populated with adjacent cavity modes, harmonic frequency combs are distinguished by phase-coherent modes that are separated by multiple cavity round-trip frequencies. Self-starting harmonic frequency combs have been discovered in both mid-IR (9−11) and THz QC-lasers (12,13) that alternate with dense comb states at different bias currents. Their intermodal spacing can be adjusted through the methods of thermal tuning, (11) optical seeding, (14) and defect engineering. (15,16) Moreover, the unique ultrafast dynamics of QC-lasers allow for an ultrafast modulation of the gain and loss, which enables harmonic mode-locking with multiple light pulses existing within a single round-trip time of the laser cavity. (17,18)

In separate experiments, THz quantum-cascade metasurface vertical-external-cavity surface-emitting lasers (QC-VECSELs) have been demonstrated as an architecture of THz QCL to achieve high-quality beam patterns, watt-level high output powers, and wide single-mode tunability. (19−21) The QC-gain material is integrated into a large-area reflective metasurface, which functions as an amplifying reflector and when combined with an output coupler, creates a resonant FP cavity. (22,23) Experiments have shown that, in contrast to ridge-waveguide QC-lasers, QC-VECSELs prefer to operate in the single-mode regime even when their gain bandwidths far exceed the longitudinal mode spacing. (24) To date, no spontaneous frequency comb generation has been observed. This is primarily attributed to the lack of spatial hole burning within the QC-VECSEL metasurface which suppresses multimode instabilities. (25) Recently, RF injection locking was demonstrated in THz QC-VECSELs. When a strong external RF signal was injected into the electrical bias of the QC-device at a frequency approximate to the cavity round-trip frequency, sideband generation was observed to induce lasing spectral broadening with a maximum bandwidth of 110 GHz. (26) The intermodal beat-note is locked to the RF injection signal, which implies that the lasing modes are evenly spaced, as a prerequisite for frequency comb/mode-locking operation.

Here, we report RF injection locking in a THz QC-VECSEL as the injection frequency is close to the integer multiples or submultiples of the cavity round-trip frequency, which is referred to as harmonic and subharmonic injection locking, respectively. Compared to the report in ref (26), an optimized QC-metasurface is used with reduced parasitic capacitance and improved RF packaging that allows for efficient harmonic injection locking up to the third harmonic; under strong RF modulation, harmonic frequency combs are triggered at resonance frequencies, where optical power is concentrated in a few modes that are spaced by several multiples of the round-trip frequency. Moreover, we demonstrated subharmonic injection locking in the same device. Taking advantage of the QC-device’s inherent nonlinearity, RF modulation at a frequency fractionally smaller than the cavity round-trip frequency is able to modulate the QC-gain and lock the intermodal beat-note. Subharmonic injection locking triggers fundamental frequency combs with comb lines spaced by one round-trip frequency, albeit with a much narrower injection locking range and frequency comb bandwidth. This process might be particularly useful for future up-scaling of the repetition rate to round-trip frequencies beyond what is directly accessible given the limits of RF packaging.

Device Design and Characterization

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The QC-metasurface used in this paper has a similar design and loaded gain material as that reported in refs (24) and (26). The active material consists of a 10 μm thick GaAs/Al_0.15Ga_0.85As heterostructure (wafer number VA 1204). Starting from the injection barrier, the layer thicknesses in Å are 51/103/17/107/37/88/37/173 (barrier layers are bold). The central 59 Å of the underlined well is Si-doped at 5 × 10¹⁶ cm⁻³. The metal−metal ridge antenna array, with a ridge width of 11.8 μm and a period of 41.7 μm, is designed to produce a broadband metasurface resonance centered around 3.4 THz with low group delay dispersion (GDD). Only a central circular area of the metasurface of 0.35 mm diameter receives current injection; such a small area reduces the power dissipation and eases continuous wave (CW) operation at 77 K. The VECSEL cavity is based upon an off-axis paraboloid (OAP) mirror with a focal length of 12.7 mm, as sketched in Figure 1a. The metasurface is further paired with a highly reflective metallic mesh output coupler (R_OC ∼ 99%) to build up an intracryostat OAP-focusing QC-VECSEL cavity with an external cavity length of 38 mm (equivalent to a cavity round trip frequency of f_rt ∼ 3.89 GHz). (24)

Figure 1. (a) Schematic of the QC-VECSEL based on an intracryostat OAP-focusing cavity design that consists of an OAP mirror with a focal length of 12.7 mm, an amplifying QC-metasurface, and a ∼99%-reflectance output coupler. (b) A picture of the QC-metasurface with an improved RF package. (c) The experimental CW P–I–V and THz emission spectrum (inset) of the QC-VECSEL collected in the free-running case. The free-running device lases in more than one lasing mode centering at a frequency of ∼3.36 THz. (d) Log scale lasing spectra and (e) corresponding beat-note collected as the RF injection frequency is swept around the cavity round-trip frequency f_RF ≈ f_rt. The red dashed box indicates the beat-notes collected in the same RF injection frequency range as the lasing spectra in (d). An injection locking bandwidth of ∼6.7 MHz is estimated and labeled in (e). (f) High-resolution FTIR spectra collected at an RF injection frequency of 3890.1 MHz. The maximum lasing bandwidth is estimated around 330 GHz.

In previous RF injection locking experiments, (26) we noted that severe RF attenuation and reflection are associated with the impedance mismatch between the 50 Ω SMA connector and the metasurface device + wire-bonding pad assembly. This in turn reduced the injection locking range and the bandwidth of spectral broadening. When compared to ridge-waveguide THz QCLs, the large area of the QC-metasurface exhibits significant parasitic capacitance and a large RC constant; in fact, the QC-metasurface is equivalent to multiple ridge waveguides in parallel, which results in a considerably smaller impedance and exaggerates the impedance mismatch. For this reason, the RF response of the device at the cavity round-trip frequency of ∼4.8 GHz in ref (26) is only 5% of that at lower frequencies, which also makes harmonic injection locking impossible. Here, we optimize the RF modulation efficiency of this device mainly in two ways. (27) First, the area of the unbiased region on the metasurface is reduced to lower its contribution of parasitic capacitance (C_MS = 1.8 pF compared to 6.1 pF in ref (26)). Thanks to the OAP-focusing cavity design, the circulating THz beam is effectively confined inside the center circular biased region with little spatial overlap with the unbiased region. This effective optical confinement allows us to reduce the unbiased area without introducing extra loss. Second, we replaced the electrical ceramic gold pad with a 50 Ω coplanar waveguide (CPW) designed on a printed circuit board (PCB) that feeds up to the edge of the metasurface chip for electrical connection to improve the RF impedance matching. Figure 1b provides a photo of the QC-metasurface with the improved RF packaging. The QC-metasurface is wire bonded to the CPW, while the latter is directly fed into a 50 Ω SMA connector that is screwed at the edge of the PCB. As a result, a microwave rectification measurement indicates the improvement of the RF modulation efficiency of this device; the 3 dB cutoff frequency is increased to 4.5 GHz and the 10 dB cutoff frequency is increased to 8.4 GHz, much better than the previous device in ref (26) (more details can be found in the Supporting Information).

The QC-VECSEL was characterized at a temperature of 77 K operating in CW mode. The free-running power and voltage vs current (P–I–V) curves are plotted in Figure 1c. The output power was collected using a pyroelectric detector (GentecEO) and is measured as 4.4 mW, after correction based on 65% transmittance of the 3 mm thick polyethylene cryostat window. It is worth noting that we intentionally introduced external optical feedback into the laser system to trigger the excitation of multiple lasing modes, which is necessary to establish an intermodal beat-note when there is no RF modulation applied (i.e., in free-running condition). The free-running spectrum, displaying more than one lasing mode, is shown in the inset of Figure 1c. This QC-device was then applied in a series of RF injection locking experiments. The experimental setup used is the same as that in ref (26). In all the following experiments, we maintained the DC bias current at 0.22 mA (≈1.44 × I_th) and the RF power at 18 dBm (nominal output level from the RF synthesizer) as we swept the RF modulation frequency. THz emission spectra and the corresponding intermodal beat-note are recorded at the same time at each RF modulation frequency. The experimental findings obtained with various RF injection strategies are discussed separately.

The RF injection signal f_RF is first swept around the cavity round-trip frequency f_rt ∼ 3.89 GHz, which we refer to as fundamental injection locking. Under strong RF modulation, lasing bandwidth broadening, and intermodal beat-note locking are observed, as demonstrated in ref (26) (Figure 1d–e). An estimated injection locking bandwidth is obtained around 6.7 MHz. It is noted that within the locking range where the greatest spectral broadening occurs, there exist regions with pedestal or side peaks that suggest an unlocked beat-note; these regions seem to be consistent with the distinct changes shown in the lasing spectra, as pointed out by black arrows in Figure 1d,e. Furthermore, the lasing spectra in Figure 1d are collected using a Fourier-transform infrared spectrometer (FTIR, Nicolet 8700) with a limited spectral resolution of 7.5 GHz, which is not able to provide precise lasing bandwidths and free-spectral ranges (FSRs) of the spectra. For this reason, we also present the high-resolution lasing spectrum collected using a home-built FTIR with a nominal resolution of 0.75 GHz. Figure 1f plots the high-resolution FTIR spectrum at f_RF = 3890.1 MHz, which clearly shows that the adjacent lasing modes are separated by one cavity round-trip frequency. The dips in the spectrum are caused by water vapor absorption along the unpurged optical path (orange dashed line). In this case, a maximum lasing bandwidth of about 330 GHz is estimated; this is significantly larger than the ∼110 GHz obtained in the previous device in ref (26); we attribute this to the optimized RF package and consequent larger modulation of the QC-gain.

Harmonic and Sub-harmonic Injection Locking

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Thanks to the improved RF modulation efficiency, beat-notes up to third-order harmonic frequency become detectable in the free-running QC-device under specific optical feedback conditions, as shown in Figure 2a. This allows us to perform harmonic beat-note injection locking─injecting an RF modulation signal that is about twice or three times the round-trip frequency, i.e., ∼7.78 and ∼11.67 GHz, respectively. (27) The collected THz emission spectra and corresponding beat-notes as a function of RF injection frequency are plotted in Figure 2. As shown in Figure 2b,e, similar lasing bandwidth broadening is observed, while the “lobes” in the low-resolution spectral maps appear to indicate an FSR increased to 2f_rt and 3f_rt compared to that in the case of fundamental injection locking. At the same time, the second and third-order electrical beat-notes clearly show pulling and locking toward the injected RF signal, with locking bandwidths estimated around 3.5 and ≤4.3 MHz─the locking bandwidth in the case of third-order harmonic injection locking is overestimated because the third harmonic beat-note signal is much weaker compared to the injected RF signal and is hard to be distinguished using the spectrum analyzer.

Figure 2. (a) Free-running beat-notes at the cavity round-trip frequency and its harmonics are detectable in the RF-improved device. The beat-notes are collected with no RF modulation applied to the QC-device. (b) Log scale lasing spectra and (c) corresponding beat-notes collected under the condition of second-order harmonic injection locking, i.e., f_RF ≈ 2f_rt. (d) The high-resolution FTIR spectrum indicates lasing modes separation of 2f_rt, where maximum lasing bandwidth around 210 GHz occurs at an RF injection frequency of 7780.1 MHz. (e) Log scale lasing spectra and (f) corresponding beat-notes collected under the condition of third-order harmonic injection locking, i.e., f_RF ≈ 3f_rt. (g) The high-resolution FTIR spectrum indicates lasing modes separation of 3f_rt, where maximum lasing bandwidth around 170 GHz occurs at an RF injection frequency of 11671.6 MHz.

Furthermore, we collected the high-resolution FTIR spectra at selected RF injection frequencies. When the RF injection frequency deviates from the second and third harmonics of the beat-note frequency, the device gives out dense spectra with an FSR of one round-trip frequency; as the RF injection frequency is close to 2f_rt or 3f_rt, a conversion between dense spectrum and harmonic spectrum is observed, where the latter one exhibits an FSR of 2f_rt or 3f_rt, respectively (see Supporting Information for all the collected spectra). In the case of second-order harmonic injection locking, the maximum lasing bandwidth is obtained at f_RF = 7780.1 MHz with an estimated bandwidth around 210 GHz, and there are around 28 detectable lasing modes with an FSR of 2f_rt; in the case of third-order harmonic injection locking, the maximum lasing bandwidth is around 170 GHz when f_RF = 11671.6 MHz with an estimated 14 modes separated by 3f_rt.

On the other hand, the inherently strong nonlinearity that arises in the QC-device allows an RF signal injected at the subharmonic frequency to lock its beat-note, which we refer to as subharmonic injection locking. (27) Three phases illustrating this method are shown in Figure 3 taking 1/2-harmonic injection locking as an example. First, an RF signal f_RF at a frequency of about 1/2f_rt ∼ 1.95 GHz is input into a spectrum analyzer which is not sent into the QC-device. Consequently, Figure 3a displays a clean RF tone, and any harmonic signals resulting from the nonlinearity of the RF instruments have been removed using an RF filter. Second, a harmonic signal of the injected RF signal is created at a frequency of about 1f_rt thanks to the nonlinearity of the QC-device when the subharmonic RF signal is fed into it. As seen in Figure 3b, this occurs even in the absence of electrical biasing and lasing activity. Third, the electrically biased QC-device produces an intermodal beat-note signal at precisely 1f_rt. It is recorded by the spectrum analyzer in addition to the harmonic signal of f_RF and can be locked to the latter (Figure 3c).

Figure 3. Illustration of subharmonic injection locking method. (a) An RF signal at ∼1/2f_rt is input into the spectrum analyzer which is not sent into the QC-device. (b) This RF signal is fed into the QC-device which is turned off. A harmonic signal at ∼1f_rt is generated due to the strong nonlinearity of the QC-device. (c) As the device is turned on, a beat-note signal at exactly 1f_rt is observed and can be locked to the harmonic signal.

The lasing and beat-note spectral maps as well as high-resolution FTIR spectra are collected as the RF injection signal is swept around half of the round-trip frequency in Figure 4a–c and one-third of the round-trip frequency in Figure 4d–f. Injection locking and lasing spectral broadening are observed but with much smaller lasing bandwidths and locking ranges when compared to the case of fundamental injection locking, despite a nearly identical amount of RF power being injected into the device. For example, under 1/2-harmonic injection locking, an estimated locking range of ∼1 MHz is obtained, with a maximum lasing bandwidth of approximately 220 GHz; under 1/3-harmonic injection locking, an estimated locking range of ∼13 kHz is obtained, with a maximum lasing bandwidth of approximately 80 GHz. It should be noticed that the spacing between each two lasing modes is kept at f_rt, meaning that dense spectra are produced under subharmonic injection locking.

Figure 4. (a) Log scale lasing spectra and (b) corresponding beat-notes collected under the condition of 1/2-harmonic injection locking, i.e., f_RF ≈ 1/2f_rt. (c) The high-resolution FTIR spectrum indicates dense lasing spectrum, where maximum lasing bandwidth around 220 GHz occurs at an RF injection frequency of 1945 MHz. (d) Log scale lasing spectra and (e) corresponding beat-notes collected under the condition of 1/3-harmonic injection locking, i.e., f_RF ≈ 1/3f_rt. (f) The high-resolution FTIR spectrum shows a maximum lasing bandwidth around 80 GHz occurs at an RF injection frequency of 1296.7 MHz.

Modeling of RF-Modulated QC-VECSEL

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In this section, we investigate the steady-state dynamical behavior of the RF-modulated QC-VECSEL using a numerical semiclassical Maxwell-density matrix approach. (28) To reduce the computational load of the simulations, we introduce an effective one-dimensional (1D) model of the metasurface and cavity geometry, where the interaction of the light field with the QC-metasurface ridge antenna is approximated by the standing wave mode of an equivalent dielectric slab. When such a model is implemented with appropriate phenomenological parameters, it reproduces the resonant QC-gain of the metasurface, including the reflective GDD. This simplification, combined with the implementation of the widely used slowly varying envelope approximation and rotating wave approximation, enables long-term QC-VECSEL simulations necessary for exploring steady-state laser dynamics. Further details of the theoretical model are provided in the Supporting Information S4.

For our simulations, we incorporate the design parameters of the amplifying QC-metasurface and OAP-focusing cavity into the Maxwell-density matrix formalism and extract the dependence of the spectral broadening by sweeping the RF injection frequency around the device’s round-trip frequency, and its harmonic and subharmonic multiples. The electric field strength of the injected RF signal is consistently set to 0.5 kV/cm for all RF injection conditions. The resulting THz emission spectra, under various RF injection scenarios, are depicted in Figure 5. These results show good agreement with the experimental observation and demonstrate a noticeable broadening of the intensity spectrum. Specifically, under harmonic injection locking, we observed a mixture of fundamental and harmonic states, while pure harmonic combs with frequency spacings of 2f_rt or 3f_rt only occur at specific frequencies, as indicated by the white arrows in Figure 5d,e. For subharmonic injections, fundamental spectra are obtained as shown in Figure 5a,b. Additionally, we observe weak beat-notes at the corresponding injection frequencies, which suggests the potential presence of additional weak modes spaced by spaced by f_rt/2 or f_rt/3. The simulated intensity distributions in the time domain are plotted at selected RF injection frequencies in Figure 5, along with the emission spectra, which show periodic intensity peaks separated by one round-trip time for fundamental and subharmonic injection locking, and two or three round-trip times for harmonic injection locking.

Figure 5. Log scale THz emission spectra and intensity distribution in the time domain simulated using a semiclassical Maxwell-density matrix method along with 1D effective modeling of the QC-metasurface under the condition of (a) 1/2-harmonic injection locking, (b) 1/3-harmonic injection locking, (c) fundamental injection locking, (d) second-harmonic injection locking, and (e) third-harmonic injection locking.

In order to investigate if the simulated optical fields are phase-locked for the cases shown in Figure 5a–e under resonant RF injection, we compute the power and phase noise quantifiers M_σP and M_ΔΦ according to ref (29). Based on the conditions that M_ΔΦ < 0.02 and M_σP < 0.005 P_av, where P_av is the round-trip-averaged optical power, we conclude that the phase-locking operation is indeed obtained for all five simulated scenarios.

Discussion

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Adler’s equation has been commonly used to characterize the phenomenon of RF injection locking with a locking bandwidth given by

Δ ν = \frac{2 ν_{0}}{Q} \sqrt{\frac{P_{i n j}}{P_{0}}}

$Δ ν = \frac{2 ν_{0}}{Q} \sqrt{\frac{P_{i n j}}{P_{0}}}$

(1)

where ν₀/Q is the cold-cavity line width, P₀ and P_inj are the powers of the free-running longitudinal mode and the injected sideband induced by RF modulation, respectively. (30,31) Adler’s equation indicates a square root dependence of the injection locking bandwidth with respect to the RF injection power. Under fundamental injection locking, we obtained an injection locking bandwidth of ∼6.7 MHz with an RF injection frequency sweeping around 3.9 GHz. As the RF injection frequency is increased from ∼3.9 to ∼7.8 GHz, the corresponding RF modulation efficiency of our device is reduced by 75% (see Supporting Information for microwave rectification results). According to Adler’s equation, the corresponding injection locking bandwidth is decreased to 3.38 MHz, that is consistent with our experimental results presented in Figure 2c. This indicates that the reduced injection locking bandwidth in the case of harmonic injection locking compared to that of fundamental injection locking is solely due to a reduced amount of effective RF power; if the same amount of RF power is applied to modulate the QC-gain, a comparably large injection locking bandwidth is expected under harmonic injection locking. This is also consistent with simulation results in Figure 5, where the same amount of RF strength applied in the simulation leads to similar lasing bandwidth under both fundamental and harmonic injection locking. On the other hand, microwave rectification measurement indicates a similar level of RF modulation efficiency at f_rt ≈ 3.9 GHz and submultiples of f_rt, which implies that the much smaller injection locking bandwidths under subharmonic injection locking are not related to the amount of injected RF power but rather originate from the introduction of an additional nonlinear process that reduces the effective RF power applied onto the QC-gain material at f_rt. The observation of subharmonic broadening and injection locking in the simulation confirms that the nonlinearity originates from the QC active material itself.

In conclusion, we demonstrate harmonic and subharmonic injection locking against an RF modulation of the electrical bias in a THz QC-VECSEL. Broadband multimode states are excited under resonant RF modulation; these results are consistent with simulation results based on a semiclassical Maxwell-density matrix approach which indicates frequency-comb operation. The phenomena of harmonic and subharmonic injection locking are similar to that reported in the case of fundamental injection locking, while the injection locking range and frequency comb bandwidth are reduced compared to the latter. In particular, the study of harmonic injection locking in QC-lasers is interesting from the point of view of both fundamental laser science and applications in, for example, arbitrary microwave waveform generation, high-speed telecommunication, as well as mid-IR and THz pulse generation. (11) The realization of harmonic frequency combs allows for the concentration of a higher amount of power into a single comb tooth, e.g., the estimated power per tooth is increased to ∼0.16 and ∼0.3 mW under second and third harmonic injection locking, compared with that of ∼0.05 mW under fundamental injection locking. As harmonic active mode-locking has been reported in ridge-waveguide THz QC-lasers, (17) our QC-device, under strong RF modulation, shows potential for operating in the active mode-locking regime with short pulse generation; further engineering of dispersion within the QC-VECSEL cavity will be needed, as well as coherent characterization techniques such as shifted-wave interference Fourier-transform spectroscopy (SWIFTs) (5,32,33) or asynchronous electro-optical sampling. (34,35) Furthermore, despite the fact that subharmonic injection locking has been realized in microwave oscillators and phase-lock loops for optical synchronization, (36) frequency division applications, (37,38) and low phase-noise frequency generation, (39) no demonstrations have been reported in QC-lasers yet. Our observation of subharmonic injection locking not only exploits the nonlinear dynamics and modulation-dependent behavior of QC-lasers, but also implies the possible integration with modern RF techniques and instrumentations, indicating promising directions for future research.

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