Dynamics of ultra-broadband terahertz quantum ... - OSA Publishing

Dynamics of ultra-broadband terahertz quantum cascade lasers for comb operation Hua Li,1,2 Pierre Laffaille,1 Djamal Gacemi,1 Marc Apfel,1 Carlo Sirtori,1 Jeremie Leonardon,3 Giorgio Santarelli,3 Markus Rösch,4 Giacomo Scalari,4 Mattias Beck,4 Jerome Faist,4 Wolfgang Hänsel,5 Ronald Holzwarth,5 and Stefano Barbieri1,* 1

Laboratoire Matériaux et Phénomènes Quantiques, Université Paris 7and CNRS UMR 7162, 10 rue A. Domont et L. Duquet, 75205 Paris, France 2 Key Laboratory of Terahertz Solid State Technology. Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning road. Shanghai 200050, China 3 Laboratoire Photonique, Numérique et Nanosciences (LP2N), IOGS - CNRS - Universités de Bordeaux, Rue F. Mitterand 33400, Talence France 4 ETH Zurich, Institute of Quantum Electronics, Auguste-Piccard-Hof 1, Zurich 8093, Switzerland 5 Menlo Systems GmbH, Martinsried, Germany * [email protected]

Abstract: We present an experimental investigation of the multimode dynamics and the coherence of terahertz quantum cascade lasers emitting over a spectral bandwidth of ~1THz. The devices are studied in freerunning and under direct RF modulation. Depending on the pump current we observe different regimes of operation, where RF spectra displaying single and multiple narrow beat-note signals alternate with spectra showing a single beat-note characterized by an intense phase-noise, extending over a bandwidth up to a few GHz. We investigate the relation between this phasenoise and the dynamics of the THz modes through the electro-optic sampling of the laser emission. We find that when the phase-noise is large, the laser operates in an unstable regime where the lasing modes are incoherent. Under RF modulation of the laser current such instability can be suppressed and the modes coherence recovered, while, simultaneously, generating a strong broadening of the THz emission spectrum. ©2015 Optical Society of America OCIS codes: (140.3070) Infrared and far-infrared lasers; (140.5965) Semiconductor lasers, quantum cascade; (190.4380) Nonlinear optics, four-wave mixing.

References and links 1.

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Received 26 Aug 2015; revised 20 Nov 2015; accepted 23 Nov 2015; published 16 Dec 2015 28 Dec 2015 | Vol. 23, No. 26 | DOI:10.1364/OE.23.033270 | OPTICS EXPRESS 33270

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Barbieri, E. Peytavit, T. Akalin, J.-F. Lampin, J. Alton, H. Beere, and D. A. Ritchie, “Metal-metal terahertz quantum cascade laser with microtransverse-electromagnetic-horn antenna,” Appl. Phys. Lett. 93, 183508 (2008). 29. A. Wei Min Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32(19), 2840–2842 (2007). 30. S. Barbieri, J. Alton, H. E. Beere, J. Fowler, E. H. Linfield, and D. A. Ritchie, “2.9 THz quantum cascade lasers operating up to 70 K in continuous wave,” Appl. Phys. Lett. 85(10), 1674–1676 (2004). 31. Water absorption dips are probably due to the fact that a small fraction (~5%) of the optical path, from the QCL to the cryostat window, was not evacuated but only purged with dried air. 32. R. W. Boyd, Non-linear optics 2nd Ed., Chapter 8, 297 (Academic, San Diego, 2003). 33. A. E. Siegman, Lasers, Chapter 6, 316 (University Science Books, Sausalito, 1986). 34. D. Walrod, S. Y. Auyang, P. A. Wolff, and M. Sugimoto, “Observation of third order optical non-linearity due to intersubband transitions in AlGaAs/GaAs superlattices,” Appl. Phys. Lett. 59(23), 2932 (1991). 35. We exclude the fact that multiple RF beatnotes are due to the presence of higher order lateral modes inside the waveguide. Indeed, using a finite element code, we found that for the present ridge waveguide higher order modes present much larger propagation losses compared to the fundamental one. We also measured the laser far field and found no clear evidence of the presence of higher order modes. 36. S. Bennet, C. M. Snowden, and S. Iezekiel, “Non-linear dynamics in directly modulated multiple-quantum-well laser diodes,” IEEE J. Quantum, Electron 33(11), 2076–2083 (1997).

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37. L. L. Columbo and M. Brambilla, “Multimode regimes in quantum cascade lasers with optical feedback,” Opt. Express 22(9), 10105–10118 (2014). 38. D. M. Kane and K. A. Shore, Unlocking dynamical diversity: optical feedback effects on semiconductor lasers, (John Wiley & Sons, Chichester, West Sussex, England, 2005). 39. In support of this hypothesis we note that for sidebands located symmetrically on different sides of the RF modulation we recorded the same broadening using the Max Hold function of the spectrum Analyzer (data not shown). This is expected since the phase noise of the RF modulation is negligible. 40. We found that when the EO crystal was perpendicular to the beam axis, changing the distance between the crystal and the QCL produced a shift of the THz modes, hence of the beat-notes in the DCE spectra. This was not the case when the crystal was tilted at 45deg, indicating a considerable reduction of the feedback. 41. J. Kröll, J. Darmo, S. S. Dhillon, X. Marcadet, M. Calligaro, C. Sirtori, and K. Unterrainer, “Phase-resolved measurements of stimulated emission in a laser,” Nature 449(7163), 698–701 (2007).

1. Introduction The exploitation of cascaded four-wave mixing for the realization of frequency combs based on micro-resonators has successfully led to the generation of octave-spanning spectra in the near-IR [1,2]. In these devices the four-wave mixing process results from the bulk third-order susceptibility (χ3) of the micro-resonator material, which eventually transforms the optical pump into a set of equally spaced comb lines. Since a few years, a different way to generate electrically pumped, chip-scale, octave-spanning frequency combs in the mid- and far-IR or Terahertz (THz) ranges is being investigated using quantum cascade lasers (QCLs) [3–5]. As for micro-resonator combs, in these semiconductor lasers four wave-mixing is believed to be the dominant mechanism leading to multi-mode operation. In this case the process of nonlinear mixing is based on the resonant excitation of the large χ(3) of the laser intersubband transition, leading to cross-gain modulation and spatial hole-burning [3,6–8]. Combined with the rather broad gain curve of QCLs, these effects lead to typical lasing bandwidths of a few THz and of a few hundreds of GHz in the mid-IR and THz ranges respectively. THz QCLs rely on two fundamentally different types active region: the so called resonant phonon (RP) active region (AR), where, as for mid-IR QCLs, the upper and lower state lifetimes of the laser transition are ruled by optical phonon emission, with values of a few ps, and the bound-to-continuum (BTC) AR, characterized by longer lifetimes up to a few tens of ps, since optical phonons play a less direct role [9]. The magnitude of the upper state lifetime is directly related to the gain recovery time, which, in turns, determines by what mechanisms the modes of the lasing spectrum can acquire a mutual coherence [10,11]. For example, in the case of BTC ARs it was shown that direct modulation of the QCL drive current at the resonator roundtrip frequency [12,13] can lead to active mode-locking, with the generation of transform-limited pulses, i.e. an equal phase-difference between adjacent modes in the emission spectrum [14–16]. Instead, in RP ARs the fact that the gain recovery time is only ~1/100 of the cavity roundtrip time has so far hindered the observation of mode-locking operation. In fact, a gain recovery time in the ps time scale has been shown to promote a phase relation between the combs modes producing a quasi-continuous wave (CW) output, or FM type of mode-locking [3]. This is in spite of the fact that, compared to BTCs, RP ARs have been found experimentally to generate significantly wider gain bandwidths (several hundreds of GHz compared to ~100GHz) and would therefore be the natural candidates for the production of short THz pulses.

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Fig. 1. Schematic representation of the three experimental setups used in this work. (a) Fourier Transform Infrared Spectrometer (FTIR). (b) Measurement setup for the RF modulation of the QCL and the simultaneous detection of the RF beatnote spectrum (see text). The black and red arrows represent respectively the modulation from the RF generator and the RF beat-note signal generated by the THz QCL. (c) Electro-optic detection of the QCL field amplitude (see text).

Even without short pulse generation, electrically pumped, octave spanning THz frequency combs would anyway represent an invaluable tool for a variety of applications, ranging from gas sensing to high resolution spectroscopy [17–19]. To this end THz QCLs with very large emission bandwidths have been reported recently, where three RP ARs at different frequencies are integrated in the same waveguide structure. The introduction of this artificial inhomogeneous broadening, combined with the broad gain transitions has allowed to achieve lasing bandwidths extending over one octave from ~1.6THz to 3.2THz [4,20]. RF beat-note measurements performed on these devices as a function of the pump current have revealed a complex behavior, with a coherent, comb-like region characterized by a single narrow beatnote signal close to the lasing threshold, followed, at higher currents, by multi-beat-notes and finally a broad beat-note spanning several GHz where the emission spectrum is the broadest [4]. Similar features, although much less pronounced, have been recently observed on a much narrower bandwidth (~200GHz) THz QCL based on a hybrid AR design [21] where resonant phonon emission should influence less the upper state lifetime, resulting into a larger value (and thus a longer gain recovery time) compared to [4]. It is interesting to observe that the above features present striking similarities to what found in the RF-spectra of microresonators combs [2]. In fact in these devices it was shown that the evolution of the optical spectrum as a function of pump power does not always lead to a set of coherent optical modes. On the contrary, it appeared that the comb coherence is strongly dependent on how the comb modes develop away from the pump, which, in turns, depends on the interplay between several parameters such as non-linear dispersion, self and cross-phase modulation and the detuning between the pump frequency and the cavity modes [2]. RF beat-note spectroscopy is a powerful tool to investigate the comb dynamics, however, even in its most sophisticated implementations such as those shown in Refs [3, 5, 22], it brings information only on the relative coherence between neighboring modes. This is a limitation in particular for broadband emission spectra, where spectrally separated groups of modes (sub-combs) can lase simultaneously as for example in [5]. For instance, as clearly shown in [2] for the case of micro-resonators combs, in this situation the observation of a single, narrow RF-beat-note signals does not preclude the fact that the different groups of modes can have different offset frequencies, hence breaking the full comb coherence. In this case, since the resolution of optical spectrometers is too low, the only way to probe the comb

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coherence is by measuring the frequencies of the different modes individually through heterodyne mixing with a tunable, single mode laser [2]. Contrary to the near-IR, in the midIR and THz ranges this is rather challenging due to the lack of sources with a sufficiently broad tuning [3,5]. In this work we circumvent this problem by using a previously demonstrated coherent sampling technique that allows probing the modes of THz QCLs spectra with unprecedented spectral resolution [14, 23, 24]. Using this technique we can measure the modes coherence and correlate it with the RF beat-note spectra, providing a deeper understanding of the dynamics of broadband THz QCLs. In particular, we investigate the broadband QCL demonstrated in [4] also subject to RF modulation. We find that the latter can dramatically improve the comb coherence while, simultaneously, considerably broaden the emission spectrum. 2. Experimental setup and basic device characterization The results presented in this work rely on three experimental setups that are schematically described in Fig. 1. For the measurement of the emission spectra of the THz QCL we used a commercial Fourier Transform Infrared Spectrometer (FTIR) with a spectral resolution of 7.5GHz (Fig. 1(a)).

Fig. 2. (a) Schematic of the mounting adopted to guide the RF modulation to the QCL (see text). The thin black lines represent the wire-bondings. (b) Photograph of the QCL, Indiumbonded to the copper holder. The microstrip-line is also visible behind the laser. The QCL image results from specular reflection on the back plane of the hyper-hemispherical Silicon lens. (c) Front view of the hyper-hemispherical Silicon lens embedded inside a copper mount. The latter allows fixing the lens in front of the QCL output facet. This is achieved thanks to a circular metal spring.

For the microwave modulation of the QCL and the measurement of the RF-beatnote spectra resulting from the non-linear mixing of the THz modes (see Section. 3.2), we used the setup shown in Fig. 1(b) [13]. An RF-synthesizer is connected through a directional coupler to a bias-tee that allows modulating the QCL, while the latter is driven at constant current using a power supply (Agilent E3642A). The RF + bias signals are brought to the QCL using an SMA cable and a 25Ω microstrip-line [12,13]. For this purpose, as shown in Fig. 2(a), the inner conductor of the SMA cable is mechanically positioned on one end of the microstripline, while the other end is wire-bonded to the QCL. One port of the coupler (labeled ‘3′ in Fig. 1(c)) is connected to a 25GHz Spectrum Analyzer (SA). This allows a continuous monitoring of the RF beatnote spectrum (red arrows in Fig. 1(b)), including the RFsynthesizer signal which, as indicated by the black arrows in Fig. 1(b), is partially reflected by the impedance mismatch at the QCL input facet. Electro-optic (EO) sampling of the QCL field amplitude is carried out using the technique described in [14,23]. The experimental setup is shown schematically in Fig. 1(c), and is based on a custom-made, harmonic mode-locked fs-fiber laser with 1GHz repetition rate and ~80fs pulse width (Menlo Systems GmbH). The THz QCL and fs-laser beams are collinearly focused on a 2mm-thick, oriented ZnTe crystal followed by two wave-plates and a polarizing beam-splitter. In this way the polarization rotation of the fs-laser beam induced by

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the THz-field is transformed into an amplitude modulation of two orthogonal polarization components that are spatially separated by the beam-splitter [23].As shown schematically in Fig. 1(c), each of them is finally sent on a custom-made balanced detection with a bandwidth of 500MHz exploiting a pair of broadband fibered InGaAs photodiodes. The balanced detection unit allows reaching the shot-noise limit by suppressing the fs-laser amplitude noise. The EO setup of Fig. 1(c) allows a coherent down-conversion of the QCL emission from the THz to the MHz range. For a detailed description of the down-conversion process we refer the reader to [14]. A short description sufficient to understand the results presented in this work will be given at the beginning of Section.3.4. The devices studied were fabricated from the same semiconductor wafer used in [4]. The AR exploits the so-called heterogeneous architecture, where different ARs are grown one on top of the other. In this case three different ARs were used, centred respectively at 2.9THz, 2.6THz and 2.3THz, for a total AR thickness of 13.12μm (see [4] and [25] for more details, and [26] for a simplified band–diagram with the important relaxation times).The common design is based on a sequence of four quantum wells. Compared to standard RP ARs [27], this design provides lower threshold current densities, thanks to a longer upper state lifetime and low free carrier absorption [26]. 50μm-wide metal-metal waveguides were fabricated using optical lithography and dry etching. In such waveguides the AR is sandwiched between a top metal layer and a bottom metallic ground-plane, which are also used as electrical contacts for current pumping. Compared to single-plasmon waveguides, metal-metal waveguides offer superior confinement properties [9], and present weaker group-velocity dispersion (GVD), which is crucial for comb operation [4]. On the other hand they suffer from a well-known drawback, i.e. a strongly non-directional far-field pattern that hinders an efficient collection and re-focusing of the emitted power [28]. To overcome this problem we have mounted a hyper-hemispherical Silicon lens in front of the laser facet (Figs. 2(b) and 2(c)) that allowed to obtain a directional beam, with a divergence of only a few degrees [29]. This step was determinant to improve the Signal to Noise Ratio (SNR) of the coherent sampling technique previously described. For the measurements, devices were cleaved into 3mm-long ridges and Indium-bonded on a copper holder (see Fig. 2(b)) that was finally mounted on the cold head of continuous flow liquid Helium cryostat. All the results reported in this work were obtained with the device operating in CW at a heat sink temperature of 30K. In Fig. 3(a) we report the Voltage vs Current and Collected power vs Current characteristics of the 50μm-wide, 3mm-long metal-metal waveguide device used in this work. The emitted power was measured with a calibrated THz power meter (Ophir 3A-P-THz ROHS), with the optical path between the QCL and the detector purged with dried air to minimize water absorption. The measured threshold current is of 340mA, corresponding to a threshold current density of 220A/cm2. In agreement with [4], above threshold the emitted power grows with a constant slope up to a maximum power of 2.2mW, beyond which it decreases due to the onset of negative differential resistance, which is typical for THz QCLs (see for example [30]). In Fig. 3(b) to 3(d) we report three representative emission spectra displayed in a normalised linear scale (see the next Section for a detailed description). For these measurements, as for all the spectra shown in this work, the FTIR was operated under vacuum to minimise water absorption.

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Fig. 3. (a) Voltage vs Current (black) and Collected power vs Current (red) characteristics of the device studied (3mm-long, 50μm-wide ridge waveguide). The laser was driven in CW at constant voltage (T = 30K). For clarity, points beyond the onset of negative differential resistance are in blue. (b,c,d) Emission spectra at difference currents, indicated by the black arrows in panel (a). Spectra were recorded with an FTIR spectrometer with a resolution of 7.5GHz. To eliminate water absorption the FTIR was operated under vacuum.

3. Results and discussion 3.1 THz emission and RF beat-note spectra in free running operation In this Section we present the THz emission spectra, measured using FTIR spectroscopy (see Fig. 1(a)), together with the corresponding RF beatnote spectra, collected using the setup described in Fig. 1(b). The QCL was operated in free running (no RF modulation). Complete colour scale intensity mappings of the THz emission and RF beat-note spectra as a function of the drive current are displayed in Figs. 4(a) and 4(b) (the range between threshold and 400mA is not displayed since there the QCL emits on a single mode at 2.7THz). The normalised THz power is reported in log scale, while the RF beat-note power is displayed in dBm. The latter was amplified using a broadband microwave amplifier with a gain of 20dB (not shown in Fig. 1(b)). Different current regions have been labelled from 1 to 6 to identify different regimes of operation. Indeed, by comparing Fig. 4(a) with Fig. 4(b) a clear correlation appears between the THz and the RF beatnote spectra. Initially, between 400mA and 420mA (region 1), we observe emission over a narrow range between ~2.7 and ~2.8THz, with a mode spacing of ~26 GHz (see also Fig. 3(b)). By comparison with measurements performed on other types of devices of the same length (see for example [13]), this value corresponds to approximately twice the expected free spectral range (FSR), indicating that one every two longitudinal modes is missing (see next Section). As a

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consequence, in this current range no RF beat-note appears in Fig. 4(b) since its frequency falls outside the bandwidth of our measurement apparatus (Fig. 1(b)).

Fig. 4. (a) THz emission (in normalized arbitrary units) vs drive current in intensity color scale. (b) Corresponding RF beat-note spectra (in dBm). The QCL was driven in CW (T = 30K) and operated in free running. The vertical dashed lines identify six current intervals corresponding to different operating regimes (see text).

Above 420mA another five different regions can be distinguished in Fig. 4, showing alternating regimes of narrow (single- and multi-peaked) and broad RF beat-notes. A few representative spectra at constant current for each of these regions are displayed in Fig. 5. In the range 420-440mA (region 2) emission is between ~2.55THz-2.85THz. The corresponding RF spectrum displays three equally spaced narrow beat-notes at 12.1GHz, 12.8 GHz, and 13.5GHz, i.e. separated by a constant value of ~0.7GHz. Narrow beat-note signals are also found in the interval 445mA-455mA (region 4) and above 470mA (region 6). In the former interval, we observe three, unevenly spaced, lines centred at 11.1GHz, 12.6GHz and 13.7GHz, while above 470mA (region 6) we find a single line at 13.3GHz. From Fig. 4(a) and Fig. 5 it appears that in these current ranges the QCL emits mainly over a few intense modes separated by wide spectral gaps. In particular a closer inspection of Figs. 5(e) and 5(i) shows that while in region 4 these modes are surrounded by a few, much less intense, additional modes, in region 6 they rather appear isolated. Finally, in Fig. 4(b) we find two regions (3 and 5) characterised by a very broad beat-note signal peaked at ~13.2GHz, and extending over several GHz. As shown in Fig. 4(a) (regions 3 and 5), and Figs. 5(c)–5(g), in this case the THz emission is broad and presents no spectral holes (the two sharp dips at 2.64THz and 2.76THz are due to residual water absorption inside the FTIR [31]).

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Fig. 5. QCL driven in free running at T = 30K. THz emission (left) and corresponding RF beat-note spectra (right) at different drive currents. The curves are extracted from Fig. 4(a), and 4(b). The red trace is the detection noise-floor. The red numbers on the left panels indicate the current regions of Fig. 4 to which each spectrum belongs.

3.2 Comparative analysis and discussion on RF beat-note and THz spectra in free running operation As pointed out in the introduction, the process of comb formation is expected be the result of four-wave mixing through the resonant excitation of the laser intersubband transition [32]. Depending on whether the process is degenerate or non-degenerate, the resulting modulation of the population inversion can lead to spatial hole-burning, with the creation of a spatial gain grating or, more generally, to cross-coupling effects, where the gain is temporally modulated at the difference frequency between two laser modes [32,33]. For both processes to take place an essential requirement is that the population dynamics is sufficiently fast. The latter is governed by the non-radiative relaxation time, τNR, which, as discussed in the introduction, is on the ps time scale. In the case of spatial hole-burning this allows the formation of a spatial gain grating, since, contrary to interband diode lsers, τNR is shorter than the in-plane carrier diffusion time [6, 7]. In the case of non-degenerate four-wave mixing the importance of a short τNR can be seen from the scheme of Fig. 6 where the process has been broken up in two consecutive steps. Initially, through laser induced saturation, the population inversion (hence the gain) is modulated at the difference frequency between two neighboring modes at ν1 and ν2. This modulation will subsequently produce two sidebands around each original mode frequency at ν1 + δ ( = ν2), ν1 - δ ( = 2ν1 - ν2) and ν2 + δ ( = 2ν2 - ν1), ν2 - δ ( = ν1). From this scheme, it is clear that the population inversion beating at δ can be efficiently generated only as long as the condition δ < 1/τNR is satisfied. In fact 1/τNR sets approximately the rate beyond

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which the response of the population inversion to en external modulation starts to be damped [34]. It was indeed demonstrated experimentally in the case of mid-IR QLCs, that resonant four-wave mixing is efficient for a detuning between the pump and signal photon frequencies up to ~1/ τNR [6].

Fig. 6. Schematic of resonant non-degenerate four-wave mixing (see text). (a) Initial mode frequencies, ν1 and ν2, separated by δ. (b) Schematic of the resonant non-linear mixing. The wavy red and black lines represent photons at ν1 and ν2. The gray line shows schematically the electronic potential profile that confines the upper and lower laser levels, represented by the horizontal black lines. As shown in the equation the population inversion Δn(t) is modulated at δ through gain saturation. E1 and E2 are the electric field amplitudes of the two modes, and ΔI(t) is the resulting current modulation. (c) Final frequencies resulting from four-wave mixing, with the two sidebands at ν1 - δ and ν2 + δ shown in green.

Concerning the non-degenerate four-wave mixing process described above it is important to observe that, to a first approximation, the current flowing through the QCL is directly proportional to the population inversion. Therefore, if a modulation of the latter is effectively taking place, it will give rise to a modulation of the current through the device as described in Fig. 6. The observation of the RF beat-notes in the spectra of Fig. 4 and Fig. 5 is precisely the evidence that a modulation of the QCL current is occurring at the difference frequency between neighboring modes of the emission spectrum, i.e. close to 13GHz for the present device [12,13]. We note that in order to help the comb proliferation process one should therefore try to enhance this current oscillation as much as possible. This can be achieved actively, through direct current modulation (see next Section), but also through the use of an adapted waveguide. To this end it was shown that compared to single-plasmon waveguides, the metal-metal waveguide used in this work presents lower propagation losses at microwave frequencies (~1dB/mm at 10GHz) and significantly higher confinement factors, thus improving the overlap between the microwave and the optical fields [12].

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Fig. 7. (a) Emission spectra in linear scale for currents corresponding to regions 1 (dashed line) and 2 (solid lines) in Fig. 4. Black and red vertical arrows indicate two sets of modes with a mode spacing of 26.3GHz (see text). Blue vertical arrows indicate two modes spaced by 2 x 26.3GHz (see text). (b) Zoom of the spectral region of panel (a) delimited by the dashed rectangle. δ1 = 12.7+/−0.3GHz and δ2 = 13.6+/−0.3GHz are the average mode intervals derived from panel (c). (c) Intervals between the modes of panel (b) for the currents belonging to region 2 in Fig. 4. Dash-dotted and dashed lines correspond to the values of δ1 and δ2 derived from Fig. 5(b), while the solid line corresponds to 13.15GHz = 26.3/2GHz.

In the remaining of this Section we will now proceed to a detailed analysis of the different beat-note regimes observed in Fig. 4 and Fig. 5. Starting from low currents, in region 2 we find three main beatnotes separated by a constant interval of 750MHz. The origin of these beatnotes can be inferred from Fig. 7, showing, in linear scale, the THz spectra belonging to region 2 of Fig. 4(a), and recorded at currents of 425mA, 430mA, 435mA and 440mA (solid lines). Black and red arrows in the Fig. indicate two sets of modes separated by a constant interval of 26.3+/−0.3GHz, i.e. equal to twice the cavity round-trip frequency. In particular, we note that the five modes indicated by the black arrows are already present in current region 1 (see the spectrum at 400mA in dashed line, Fig. 4 and Fig. 3(b)), whereas the two modes under the red arrows appear only in region 2, together with the three evenly spaced beatnotes. In Fig. 7(c) we report the values of the mode spacings deduced from the frequencies of the four peaks inside the dashed rectangle of Fig. 7(a). Within the errors produced by the limited accuracy (~250MHz) of the FTIR spectrometer, at all currents we observe two distinct spacings. Their average values are δ1 = 12.7+/−0.3GHz and δ2 = 13.6+/−0.3GHz, in very good agreement with the frequencies of the two most intense beat-notes of Fig. 5(b), δ1 = 12.81GHz and δ2 = 13.55GHz (dash-dotted and dashed horizontal lines in Fig. 7(c)). As shown in Fig. 7(b), δ1 and δ2 are alternated, therefore

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preserving the constant interval of 26.3GHz between two consecutive black or red arrows. As a result the modes indicated by the black and red arrows belong to two distinct sub-combs characterized by different offset frequencies. The difference between the offsets is given by δ2-13.15GHz (26.3GHz/2). Clear evidence of sub-combs sharing a common spacing but with different offset frequencies was found in micro-resonator combs based on four-wave mixing [2]. It was shown that their formation depends on the interplay between GVD, self and cross-phase modulation and the detuning between the pump frequency and the cavity modes. Here, we believe that essentially the same type of mechanism is taking place. At 400mA, the doublespaced comb indicated by the black arrows in Fig. 7(a) is produced by four-wave mixing starting from mode B and generating modes A to F (see Fig. 6 and the description and the beginning of this Section). Next, at 425mA, modes G and H appear, together with modes M and N. As shown above, modes G and H, despite being separated by twice the roundtrip frequency, do not belong to the original double-spaced comb. We attribute this to the effect of GVD, yielding a different group index for mode G compared to mode B. In turns, mode H is generated from mode G thanks to the fact that a current modulation at 26.3GHz is taking place in the QCL through the mutual beating of the modes under the black arrows: this beating generates mode H as a sideband of mode G. The same type of process gives rise to modes N-M, which are separated by 2 x 26.3GHz (as for mode G, mode N is not separated from mode B by an integer multiple of 13.15GHz = 26.3GHz/2)

Fig. 8. Schematic of the generation of the lowest frequency beat-note in the spectrum of Fig. 5(b). By exploiting the intrinsic non-linearity of the laser Current-Voltage characteristic a first beating (black cross labeled “1” in the Fig.) is produced at Δ = δ1 - δ2. This frequency is finally mixed (black cross labeled “2” in the Fig.) with δ1 to produce a third line at δ3 = δ1 - Δ.

The above picture shows that the mode formation process can follow an intricate dynamics, resulting from four-wave mixing, GVD and current modulation through mutual mode-beating. As a consequence the mode spacing does not necessarily reflect the local value of the group index (for instance the spacing between modes E and G is not imposed by the value of the group index at ~2.8THz) [35]. From the ~500MHz frequency shift of mode G with respect to 7 x 13.15GHz (see Fig. 7(c)), one can only derive an average GVD of ~1 x 105 fs2/mm in the range 2.7-2.8THz (see Appendix 5.1). This value is compatible with the GVD evaluated numerically in [4]. The lowest frequency beat-note of Fig. 5(b) is instead not generated by the direct beating of two THz modes, but is the consequence of the intrinsic non-linearity of the QCL CurrentVoltage characteristics. As shown schematically in Fig. 8, its generation takes place in two steps. First the two highest frequency beatnotes, oscillating at δ1 and δ2, are mixed by the device electrical non-linearity, producing their difference frequency, Δ. Next Δ beats with δ1,

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generating δ3 = δ1 -Δ (two additional sidebands at frequencies δ1 ± 2Δ are also visible in Fig. 5(b)).

Fig. 9. (a) Low frequency part of the spectrum of panel (b). The spectrum was obtained by subtracting the background in order to remove spurious RF lines. (b) Same RF-beat-note spectrum of Fig. 5(b). (c) Spectra of the beat-notes oscillating at δ1, δ2 and δ3 in panel (b) obtained by operating the Spectrum Analyzer in the Max. Hold mode during 3 minutes. The frequency offsets were removed. (d) Result of a two-tone modulation of the QCL current at 11.8GHz and 13.4GHz (see text).

In Fig. 9 we report further experimental evidence supporting this interpretation of the data. Figure 9(a) shows the low frequency end of the RF spectrum of Fig. 9(b) (same as Fig. 5(b)). Here the intense line at 750MHz corresponds to Δ = (δ2-δ1), demonstrating that the beat-notes of Fig. 9(b) are indeed separated by a constant spacing. In Fig. 9(c) we report the three most intense beat-notes of Fig. 9(b) recorded during a time interval of 3 minutes with the Spectrum Analyser operated in the Max. Hold mode (the beat-notes center frequencies are subtracted for clarity). We find that the lowest frequency beat-note (blue curve) presents a frequency-noise induced broadening that roughly equals three times the broadenings of the center and highest frequency beat-notes (black and red curves respectively). From the schematic of Fig. 8, this is expected since δ3 = δ1 -Δ = 2δ1 – δ2, and the frequency drifts of δ1 and δ2 are, to a first approximation, uncorrelated. Finally, Fig. 9(d) shows a spectrum recorded with the QCL operated just above threshold (at this current no RF beat-notes are observed). Here the two most intense lines are the result of a two-tone modulation [36] of the QCL current at frequencies of 11.8GHz and 13.4GHz obtained using two separate synthesizers. As a consequence of the intrinsic non-linearity of the Curent/Voltage characteristic of the device, these two frequencies produce precisely the same effect observed in Figs. 9(a) and 9(b), that is the generation of a lower frequency beat-note at their difference frequency of 1.6GHz (including its second harmonic), which, in turn, generates a lower modulation sideband at 10.2GHz.

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Overall, the results presented in Fig. 9 highlight the fact that the non-linear process based on electrical rectification described in Fig. 8 can as well play a role in the generation and proliferation of sidebands in the optical range. This process is in fact equivalent to a microwave four-wave mixing, and in the absence of GVD could favor the formation of harmonics of a given comb spacing (in this case δ1 and δ2 would be respectively the comb spacing and its second harmonic). In Figs. 5(e),5(f),5(i),and 5(l), corresponding to current regions 4 and 6, we observe respectively the presence of three and one single narrow RF beat-notes. The corresponding THz spectra present small groups of modes or a few scattered modes, separated by wide spectral gaps. Due to the fact that neighboring modes have very different intensities, and owing to the spectral resolution of the FTIR spectrometer limited to 7.5GHz, it was not possible to determine the modes spacing with sufficient precision to allow attributing the observed RF beatnotes to specific spectral regions, as was done for the spectrum of Fig. 5(b). As shown in Fig. 4(b), current regions presenting multiple RF beat-notes, are separated by regions characterised by a single broad beat-note spanning a few GHz. The transition from a narrow to a broad beat-note regime comes together with an analogous transition in the THz spectrum (Fig. 4(a) and Fig. 5), namely from a regime where the emission spectrum consists of scattered groups of modes separated by fairly broad spectral gaps, to a regime where (i) virtually no modes are missing, and (ii) the spectrum has expanded towards lower frequencies (see in particular Fig. 5(g)). Abrupt transitions from narrow to broad beat-note regions as a function of the pump current such as those observed in Fig. 4(b) have been found also in [4] for the same AR used in this work, and in [21], for a QCL based on a single stack AR, and emitting on a much narrower spectral range, from approximately 4.1THz to 4.3THz. This last finding shows that this phenomenon is not restricted to multi-stack QCLs. As will be shown in the next Section, the appearance of a broad RF beat-note when the emission spectrum becomes strongly multi-mode is the result of the QCL entering an unstable regime of operation where the modes lose their mutual coherence (see next Section). The origin of this phenomenon is an open question and goes beyond the scope of this work. Recently, it was predicted theoretically that at high levels of optical feedback, single-mode mid-IR QCLs may display stable or chaotic multi-mode operation depending on the feedback coupling [37,38]. So far no such studies have been performed on multi-mode THz QCLs. In this work, as was the case for [4] and [21], we found that the RF beat-note spectra displayed in Fig. 4(b) could indeed be heavily modified when the QCL was subject to strong optical feedback, such as obtained by focusing the laser on a metal mirror (see also Section 3.4). To rule out any significant effect of feedback on the spectra of Fig. 4(b), which were collected with the QCL emitting inside the FTIR spectrometer, we compared them to analogous spectra obtained when the laser was emitting in open space, i.e. without any optical element across the beam with the exception of the cryostat window (the hyper-hemispherical silicon lens described in Section 2 was also removed for this test). We did not observe any significant change, showing that those found are genuine features resulting from the multimode dynamics of the free running laser. 3.3 THz emission and RF beat-note spectra under RF modulation In Figs. 10(a) and 10(b) we report colour scale intensity mappings of the THz and RF beatnote spectra as a function of the drive current. Spectra were recorded with the QCL current modulated at 13.22GHz, and with + 23dBm of RF power from the synthesizer. A few representative spectra at constant drive current, derived from Figs. 10(a) and 10(b), are also reported in Fig. 11. From the experimental setup described in Fig. 1(b), we recall that in the measured RF beat-note spectra we observe simultaneously the reflected RF modulation signal and the beat-note(s) generated by the mixing of the various THz modes. We also note that, due to the attenuation of the RF cables, directional coupler and bias-tee (Fig. 1(b)), as well as the unavoidable reflection/radiation of the RF modulation signal at the interface between the

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SMA cable and the QCL (Fig. 2(a)), we estimate a total power loss of ~20dB from the RF generator to the QCL. This yields a current modulation of ~10mA amplitude for + 23dBm of RF power at 13.2GHz. Compared to operation in free running (Fig. 4) both the THz and RF spectra are dramatically modified. In particular we find two main features: (i) THz emission is significantly broadened and presents no spectral holes (except for the two sharp water

Fig. 10. (a)T Hz emission (in normalized arbitrary units) vs drive current in intensity color scale. (b) Corresponding RF beat-note spectra (in dBm). The QCL was driven in CW (T = 30K) and modulated at 13.22GHz with a synthesizer power level of + 23dBm. The vertical dashed lines identify three current intervals corresponding to different operating regimes (see text).

absorption dips at 2.64THz and 2.76THz also observed in Fig. 4) [31]; (ii) the intense and broad RF beatnotes found in free running have been strongly reduced. These features remain virtually unchanged when the value of the modulation frequency is moved inside an interval of several hundreds of MHz from 13.2GHz (not shown). For larger detunings the phase-noise suppression was found to be less pronounced, but this point needs further investigations. As shown in Fig. 10 three current regions can be identified. For currents up to 450mA (region 1), the three narrow beat-notes and the broad beat-note found in Fig. 4 have disappeared, and are replaced by sets of low-intensity, unevenly spaced, narrow sidebands, positioned symmetrically on both sides of the modulation frequency. Examples of these sidebands are reported in Figs. 11(b), 11(d), and 11(f), and we note that their intensity is at least 20dB below the most intense sidebands of Fig. 4. They could originate from the same type of process shown in Fig. 8, involving the mixing of the external modulation frequency with the beat-note between two neighboring comb modes or of a group of modes sharing the same frequency spacing [39]. However in this case we would expect an asymmetric distribution of the sideband intensities, as found in Fig. 5(b). An alternative explanation is based on the same type of process described in Fig. 7, where two groups of modes sharing a common spacing but having different frequency offsets are partially overlapped [2]. This time the common mode spacing of the two sub-combs would be dictated by the external RF modulation. When the tails of the sub-combs are spectrally overlapped, RF sidebands are produced on both sides of the common mode spacing, at a distance given by the offset frequency difference (see the next Section). A description of this process is given in Appendix 5.2. We have tried to identify the presence of sub-combs by deriving the mode spacings from the measurement of the position of the THz modes in the emission spectra. However, owing to the intrinsic error given by the limited accuracy and spectral resolution of our spectrometer, we could not find any clear evidence.

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Fig. 11. QCL modulated at 13.22GHz with a power from the synthesizer of + 23dBm (T = 30K). THz emission (left) and corresponding RF beat-note spectra (right) at different drive currents. The curves are extracted from Fig. 10(a), and 10(b). In the RF spectra the peak of the 13.22GHz line is at −25dBm, and corresponds to the intensity of the RF modulation signal reflected by the device (Fig. 1(b)). The intensity scale was deliberately reduced in order to magnify the effect of the phase noise. The red trace corresponds to the detection noise-floor. The red numbers on the left panels indentify the current regions of Fig. 10 to which each spectrum belongs.

At higher currents (region 2, Fig. 10) we observe the appearance of a broad phase noise pedestal around the modulation frequency. In particular, a comparison between the left and right columns of Fig. 11 shows very clearly that the latter appears as soon as the QCL starts to emit below ~2.5THz. By further increasing the current (region 3) the intensity of the phasenoise pedestal is increased, and we assist to a further expansion of the low-frequency end of the THz spectrum. This could be the signature of an increasing GVD below ~2.5THz compared to higher frequencies [4]. By comparing Fig. 11 with Fig. 5 we note that the phasenoise pedestal in the presence of RF modulation remains always at least 30dB below the intensity of the broad beatnote observed in free running: in general, regardless of the pump current, the RF modulation is always effectively improving the mutual coherence of the THz modes. This issue will be studied in more detail in the next Section with the help of EO sampling.

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3.4 Probing the coherence of the THz modes with electro-optic detection The EO sampling technique based on the experimental setup of Fig. 1(c) allows a coherent down-conversion of the QCL emission from the THz to the MHz range [14]. The principle of the down-conversion process is illustrated schematically in Fig. 12(a) for three arbitrarily spaced THz modes. As shown in the Fig., each QCL mode frequency ν is transformed into a frequency given by f = |ν – n x f repfs |, where f repfs is the fs-laser repetition rate and n is an integer such that n x f repfs is the closest harmonic of f repfs to ν. As a result the full emission spectrum is down-converted in the spectral region between dc and f repfs /2 = 500MHz. As shown schematically in Fig. 12(a), this process does not in general preserve the original modal ordering. The fundamental reason for this is that the optical sampling of the QCL emission using the fs-pulse train is by far below the Nyquist rate [14].

Fig. 12. (a) Schematic representation of the down-conversion process produced by the EO sampling of three arbitrarily spaced modes of frequencies ν1, ν2 and ν3. The THz frequencies are coherently transformed into the corresponding frequencies f1, f2 and f3 in the MHz range (see text for the explanation).(c) Schematic of the down-conversion of a THz QCL comb with a mode spacing

QCL f rep (see text).

Despite this inconvenience, the great advantage brought by this technique is that each down-converted beat-note signal preserves the phase-coherence of the original THz mode. Indeed, as was demonstrated in [24] the phase (frequency) noise of n x f repfs is two orders of magnitude below that of the generic QCL mode in the 2-3THz range, therefore the phase noise of f = |ν – nx f repfs | is equal, with a very good approximation, to the phase noise of ν. In other words, the fs-laser can be used as a multi-line local oscillator to measure the coherence of the THz QCL modes with a very high spectral resolution (essentially the resolution of the RF spectrum analyser shown in Fig. 1(c)). For completeness we note that in the case where QCL ), the QCL spectrum is that of a comb (i.e. with evenly spaced modes separated by f rep provided that the difference frequencies between the comb lines and the same harmonic of f repfs are all below f repfs /2, then the relative positions between the modes is preserved. As shown schematically in Fig. 12(b), in this particular case one would obtain an RF replica of

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QCL the original THz comb, with a mode spacing given by | f rep – n x f repfs |, where n x f repfs is the QCL [14]. closest harmonic of f repfs to f rep

Fig. 13. Examples of DCE spectra (left column) and their corresponding THz emission spectra (right column) at different currents. The DCE spectra were collected with a resolution bandwidth of 1MHz. The spectral resolution of the emission spectra is 7.5GHz. The red dashed lines in the right column represent the minimum THz intensity detectable with the EOS system as deduced from the spectra in the left column. In panels (a), (b), (c), and (d) the QCL was operated in free running. In panels (e) and (f) it was RF modulated at 13.2GHz with + 24dBm from the synthesizer.

In the left column of Fig. 13 we report a few examples of down-converted emission (DCE) spectra of the QCL, obtained with the EO sampling setup of Fig. 1(c). The corresponding THz spectra collected with the FTIR spectrometer (Fig. 1(a)) are displayed in the right column. We stress that although both sets of spectra were measured at the same drive currents and same heat sink temperature (30K), they could not be collected simultaneously. This is a relevant issue since when performing EO sampling the QCL was unavoidably subject to feedback resulting from the reflection of the THz beam on the EO crystal, which, as discussed in Section 3.2, can modify the THz emission and RF beat-note spectra [24]. However, by tilting the EO crystal at ~45deg with respect to the beam axis, we minimized the feedback, and verified that in this condition the RF beat-note spectra were not significantly distorted compared to those displayed in Fig. 4 and Fig. 10 [40]. By comparing the two sets of spectra reported in Fig. 13 it is apparent that, as explained above, the relative position of the THz modes is not preserved in the down-conversion process. On the other hand the linewidth of the down-converted THz modes is not anymore limited by the instrumental spectral resolution. In particular we find that the average linewidth of the THz modes, which in the right column of Fig. 13 is of 7.5GHz (the FTIR spectral resolution), is reduced to only a few MHz in the DCE spectra. As discussed in [14, 23, 24] such “instantaneous” linewidth corresponds to the slow frequency drift of the QCL modes

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determined by technical noise, i.e. current, thermal and mechanical fluctuations (this same fluctuations give rise to the drifts observed in Fig. 9(c)). At this point a very important remark is that the observation of drift-limited linewidths in the DCE spectra, i.e. drift-limited THz modes, was always found to lead to narrow (single or multiple) beat-notes in the corresponding RF spectra and vice versa. Indeed the DCE spectra of Figs. 13(a) and 13(c) correspond to regions 2 and 6 of Fig. 4(b) while the RF beat-note spectrum relative to Fig. 13(e) is displayed in Fig. 14(g). These are only particular examples of a generally verified trend. As we shall see below, the situation changes when the RF beat-note presents a broad phase-noise pedestal.

Fig. 14. (a) RF beat-note spectra and (b) corresponding DCE spectra vs RF modulation power in intensity color scale. The QCL was driven in CW at a current of 450mA (T = 30K). The vertical dashed lines identify three regions corresponding to different operating regimes (see text) (c) RF beat-note spectrum without RF modulation. (d) Corresponding DCE spectrum. (e),(g), and (i): RF beat-note spectra extracted from panel (a). The red numbers indentify the regions of panels (a) and (b) to which each spectrum belongs. (f),(h), and (l): corresponding DCE spectra extracted from panel (b). The red curves represent the noise floor. Note that the spectrum in panel (h) is the same to that of Fig. 13(e).

In Figs. 14(a) and 14(b) we report sets of RF beat-note spectra and their corresponding DCE spectra in intensity color scale for an increasing RF modulation power, from 0 to + 26dBm. The QCL was driven at 450mA, and without RF modulation presents a broad RF beat-note (Fig. 14(c)). As shown in Fig. 14(a), by switching ON the modulation at 13.22GHz and increasing the modulation power we find a progressive reduction of the RF beat-note width, until, between + 23 and + 25dBm, the phase-noise goes below the detection noise floor. In the corresponding DCE spectra of Fig. 14(b), which were measured simultaneously, we find that as long as the phase-noise pedestal remains above the detection noise floor, no narrow lines can be observed. Instead we find a weak and spectrally broad signal, only a few dB above the detection noise floor, as shown also in Fig. 14(d) and 14(f). This implies that the THz modes are completely incoherent, i.e. their linewidth is not limited by technical noise, but is instead broadened by the same instability found in Section 3.2 without RF modulation. Since the RF beat-note results from the beating between adjacent THz modes, the fact that its phase noise pedestal extends over more than 1GHz is consistent with the fact that the RFlines in the DCE spectra are completely washed out. Indeed we recall that (i) the phase-noise of the THz modes is faithfully reproduced in the DCE spectrum, and (ii) the frequency of each down-converted mode falls necessarily in the range 0-500MHz ( = f repfs /2. See Section #248434 © 2015 OSA


2). As shown in Fig. 14(d), even when the QCL is not modulated, we find the same correlation between the broadening of the RF-beat-note and the complete loss of coherence of the THz modes. We note that the same type of effect was observed in [21] on a free running QCL, using a high resolution (100MHz) FTIR spectrometer. The THz modes recover their coherence (i.e. their drift limited linewidth) when the RF beat-note phase noise disappears below the detection noise floor. This is shown in Fig. 14(b), where the DCE spectra display a number of sharp lines for RF powers between + 23dBm and + 25dBm, producing an attenuation of the peak of the broad RF-beat-note by more than ~30dB (compare Fig. 14(c) with Fig. 14(g)). As an example, in Fig. 14(h) (same as Fig. 13(e)) we report a DCE spectrum measured with a synthesizer RF power of + 24dBm. Finally, by increasing the RF power beyond + 25dBm the THz modes broaden again, as shown in Fig. 14(l). The reason for this loss of coherence at high RF power is still unclear. The observed abrupt effect of induced coherence by an external RF modulation was never observed before in THz QCLs. Indeed previous experiments demonstrating RF injection locking of the inter-mode beating [13] were performed on devices based on BTC ARs (see Section 1), which, contrary to RP ARs, always displayed narrow inter-mode beatings (to the best of our knowledge, broad intermode-beatings have been observed only in the present AR [4] and in the one reported in [21]). Therefore, in that case, the effect of RF-injection locking was solely that of suppressing the low frequency drift of the inter-mode beating due to technical noise. At this point one could ask whether the abrupt recovery of the THz modes coherence induced by the RF modulation found in this work is as well the result of an RF injection-locking. Indeed, the progressive reduction of the intensity of the phase-noise pedestal observed in Fig. 14, followed by a sudden vanishing of the latter at ~ + 23dBm would be compatible with this type of process. In fact we recall that when increasing the RF power at a fixed modulation frequency, the typical dynamics of an RF injection-locking process consists of (i) an initial pulling of the free running inter-mode beating, followed by (ii) an abrupt injection [13]. During the pulling phase only the line intensity is reduced, while its linewidth remains virtually unaffected. The latter eventually collapses to that of the RF generator as soon as the beat-note enters the locking range (in the present case the locking range would be roughly given by the width of the phase-noise pedestal at + 23dBm RF power, i.e. of ~250MHz (see Fig. 14(a)). Another interesting finding from the data of Fig. 14 is that no coherent modes appear while increasing the RF power from 0 to + 23dBm. In fact this behavior definitely rules out the simple picture where the broad phase-noise pedestal in the RF beat-note is due to the superposition of independent beatings between adjacent THz modes. Indeed, in this case we would expect the appearance of an increasing number of narrow lines in the DCE spectra, resulting from an increasing number of beatings being injection-locked by the external RF modulation. Instead, the observation of an abrupt transition from incoherent to coherent modes suggests that the RF-beat-note broadening is rather the result of a collective instability involving all the modes simultaneously. Further support to this interpretation comes from the data presented in Fig. 13, which, again, provide evidence that coherent and incoherent THz modes cannot coexist simultaneously. To this end the horizontal dashed red lines in the Fig. reproduce the minimum intensity detectable with EO sampling as obtained from the DCE spectra (i.e the ratio between the intensity indicated by the red line and the most intense THz mode, is equal to the ratio between the noise floor of the corresponding DCE spectrum and the intensity of the most intense RF-line). By counting the number of THz modes above the red dashed lines and the number of RF-lines we find that they are exactly equal in all the displayed spectra, implying that within the SNR of the EO sampling the THz modes are all coherent. In particular when the QCL is modulated (Fig. 13(e) and 13(f)) it means that within a dynamic range of ~16dB (a factor of 40 in linear scale) all the THz modes have recovered their drift limited coherence.

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Fig. 15. DCE spectrum of the QCL driven in CW at a current of 450mA. The resolution bandwidth is 1MHz and the Spectrum Analyser is operated in Max. Hold mode. The laser is RF modulated at 14GHz, with a power of + 24dBm from the synthesizer. The lowest frequency line (blue arrow) is phase locked to the fs-laser repetition rate (see text). The red arrows indicate the lines that are not drift-broadened.

A rigorous way of establishing whether the modes of a THz QCL are RF-injection locked was demonstrated in [14] by exploiting the same EO sampling technique used in this work, with the additional requirement that one of the THz modes must be phase-locked to the repetition rate of the fs-laser (see [14]and Appendix 5.3 for experimental details). In fact, in this case, if the QCL inter-mode spacing is RF injection-locked, all the down-converted THz modes in the DCE spectrum should display a zero frequency drift, since they would all be phase-locked to the fs-laser repetition rate [14]. In Fig. 15 we report an example of DCE spectrum obtained with the THz QCL under the effect of RF modulation at 14GHz and + 24dBm of RF power (the corresponding RF beatnote spectrum, not shown, is similar to the spectrum of Fig. 14(g) and presents no phase-noise pedestal). Here, following the technique of [23], the lowest frequency RF-line at ~44MHz (blue arrow) was used to phase-lock the corresponding THz mode to the harmonic of the fslaser repetition rate (see the Appendix for experimental details). The spectrum was collected with the Spectrum Analyser in the the Max. Hold mode during approximately 30sec. Surprisingly, we clearly observe that the RF-lines present different drift-induced broadenings. In particular we can identify four lines, indicated by the red arrows in the Fig., with a linewidth equal to that of the phase-locked beat-note (blue arrow), i.e. limited by the analyzer resolution bandwidth (1MHz). Definitely, these five lines correspond therefore to THz modes that have been made phase coherent by the RF modulation. The remaining beat-notes are instead significantly broadened by frequency drifts. We stress that in all the DCE spectra recorded in the same conditions of Fig. 15, i.e. under simultaneous phase-locking and RF injection locking but at different pump current and RF power levels, we always found the same result: a significant number of lines were subject to a frequency drift resulting from technical noise. In light of the above findings there are two possible scenarios. The first possibility is that RF modulation locks the inter-mode beating of only a subset of THz modes, while, at the same time forcing the remaining others to recover a drift-limited linewidth. Indeed, as shown in Fig. 14(d), we recall that without RF modulation the modes are completely incoherent. Given the dynamics of RF-injection locking described earlier, namely that during the pulling

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phase the inter-modal beating line-width remains unaffected, this behavior seems to us hard to justify. Another possibility is that the THz emission spectrum is composed of sub-combs with the same inter-mode spacing but with different frequency offsets, following the same type of mechanism leading to the spectra of Fig. 7. In this case, an RF-injection locking that would mutually lock neighboring THz modes across the entire emission spectrum would still be compatible with the observation of the drift-broadened RF lines in the DCE spectrum of Fig. 14. Indeed the phase-locking of the 44MHz RF line would lock the offset of only one subcomb, hence freezing the drift of the THz modes belonging to it thanks to RF-injection (these modes would give rise to the five narrow lines indicated by the red arrows). On the other hand the offsets of the remaining sub-combs would still be free to drift independently, thus giving rise to jitter-broadened RF lines in spite of RF-injection. Further studies will be needed to prove which of the two scenarios is correct. 4. Conclusions We have studied the coherence dynamics of octave spanning THz QCLs [4] using RF beatnote spectroscopy and EOS. Although the data presented in this work are relative to a single, 50μm-wide, 3mm-long metal-metal waveguide device, we stress that we studied several other devices and found a qualitatively similar behaviour (see also [4]). Namely, depending on the drive current, we observe different regimes ranging from stable multimode emission to incoherent multimode dynamics [4]. In the first case the measured RF beat-note spectra display single or multiple narrow lines. From the analysis of the emission spectra we have shown that these lines can result from the beating of THz modes belonging to sub-combs with different offset frequencies. The formation of these sub-combs is ultimately triggered by GVD, and is the result a complex dynamics involving four-wave mixing and sideband generation through the mutual beating of THz modes. Incoherent emission is observed each time the laser emits on a broad spectral range (~2.1-3THz) with virtually no spectral holes. In this case the RF spectra display a single beat-note characterized by an intense phase-noise, extending over a bandwidth up to a few GHz. Using EOS we find that in this case the QCL enters a regime of unstable operation, where the THz modes lose their otherwise high level of coherence, essentially determined by technical noise. The onset of this phenomenon is abrupt and its origin must still be clarified. Under the effect of an external RF modulation centered at the peak of the broad RF beatnote (~13.2GHz) the QCL dynamics is radically modified. In particular by increasing the RF power the broadband phase noise is progressively reduced until the THz modes abruptly recover their technical noise-limited coherence. At the same time we observe a significant broadening of the emission spectrum. To the best of our knowledge such RF-induced coherence was never observed before in QCLs. Our measurements show that this process is compatible with RF injection locking, where the external RF modulation compensates the effect of GVD and locks the frequency difference between neighboring THz modes. However we have no clear evidence that this produces a THz frequency comb of evenly spaced modes. Indeed measurements using EOS show that only a limited number of THz modes are mutually phase-locked. At the moment, in analogy with what found at lower currents when the QCL beatnote spectrum displays multiple narrow lines, we speculate that this is the result of an emission spectrum composed of groups of modes sharing a common, RF-injected, mode spacing but having different offset frequencies because of GVD [2]. The results presented in this work show therefore that the compensation of GVD remains a crucial step for the achievement of a QCL-based, broadband THz frequency comb [5]. To this end we note that an accurate measurement of the dispersion of the present laser using techniques such as THz time domain spectroscopy is still lacking [5,20,41]. At the same time our measurements clearly show that RF modulation remains a key ingredient to control the comb coherence and, in particular, to suppress the regions of incoherent dynamics. In this context, it is clear that understanding the physical origin of the observed incoherent emission

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could bring new insights on how to favor frequency comb operation, besides constituting an interesting fundamental problem on its own right. To this end, measurements of the QCL ouput vs time are currently under way using a fast Schottky diode (see also Appendix 5.2). 5. Appendix 5.1 Evaluation of the GVD From Fig.7, by taking δ ∼ 13.15GHz (= 26.3/2 GHz) as the laser roundtrip frequency in c ≈ 3.8 , where proximity of mode B at 2.73THz, we obtain a group effective index nGeff = 2δ ⋅ L L= 3mm is the cavity length and c the speed of light. From Fig.7(c), the shift of mode G with respect to 7δ (i.e. the frequency difference between mode G and mode B) is given by neff Δ ≈ 500 MHz, yielding a change of effective index ΔnGeff ≈ G Δ = 0.02. ΔnGeff represents the 7δ average change of effective index in the spectral region between mode B and mode G, that is in a bandwidth f B ≈ 90 GHz (7δ), from 2.7THz to 2.8THz. Hence we derive an average GVD  ΔnGeff 1 in this interval given by   c 2π ⋅ f B  GVD evaluated numerically in [4].

  ~1x 105 fs2/mm. This value is compatible with the  

5.2 Independent sub-combs As discussed at the end of in Section 3.3 a possible explanation for the appearance of equally spaced spectral lines in some of the RF beat-note spectra of Fig.11 is that the emission spectrum is composed of two (or more) independent sub-combs with different frequency offsets but sharing the same inter-mode spacing [2]. This process would follow precisely the same dynamics leading the two highest frequency beatnotes of Fig.5(b) and described in Section 3.2 (see Fig.7). This time, however, the comb modes are separated by ~13.2GHz, rather than 26.3GHz. As a result two additional beatnotes are observable, which were previously outside the RF detection bandwidth. As schematically shown in Fig.16(a)–(c) when the tails of two sub-combs with different offsets overlap spectrally, two sidebands should appear in the RF beat-note spectrum (Fig.16(c)) on both sides of the common spacing, labeled δ. The frequencies of these sidebands are given by δ ± Δ , where Δ is the difference between the offset frequencies. A beat-note at Δ should also be observed in the low frequency end of the RF spectrum [2]. In Fig.16(d) and 16(e) we report an example of measured RF beat-note spectrum displaying precisely these features (see also Fig.9(a) and 9(b)). In this case we find two different sets of sidebands which should indicate the presence of three groups of modes. However, as explained in Fig.8, it is important to note that the appearance of the two sidebands at low frequency is likely to be the result of the non-linearity of the QCL Current-Voltage characteristic, producing the beatings between the RF lines in Fig. 16(e), rather than a direct mixing of the THz field as shown in Fig.16(b) and 16(c). Therefore the spectrum of Fig.16(e) cannot be unequivocally interpreted as a signature of the existence of the sub-combs.

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Fig. 16. (a),(b),(c)Schematic of the RF beat-note generation by two groups of independent subcombs with different frequency offsets and sharing the same inter-mode spacing, δ. Δ is the frequency difference between the offsets. The black cross indicates the mixing process inherent to a process of quadratic (power) detection. (d),(e) Example of experimental RF multi-line beat-note spectrum that could be interpreted as the signature of the presence of subcombs.(f),(g) Example of broad RF beat-note spectrum. The low frequency spectraof panels (d) and (f) were obtained by subtracting the background in order to remove spurious RF lines.

To answer this question one should measure the QCL RF spectrum using a fast external power detector, such as a Schottky diode, operated in a linear regime. This would be relevant also to study the regime of broad beat-note. Indeed, in the case of incoherent multimode emission we expect to observe a random fluctuation in time of the output power, which could be measured using a fast detector. In particular, when the latter is connected to a SA, such fluctuations should give rise to the appearance of broadband noise close to dc, generally referred to as amplitude noise [2]. As shown in Fig. 16(d) and 16(e) (and Fig. 9(a) and 9(b)) in the presence of narrow RF beat-notes we find no additional noise at low frequencies. Instead when the RF-beat-note is broad (Fig. 16(f) and 16(g)), we observe the appearance of a broadband, low-frequency noise. If the absence of low frequency noise (Fig. 16(d)) is indeed a signature that the QCL output power is regular in time, on the other hand, again owing to the QCL electrical non-linearity, the observation of broadband noise cannot be unequivocally interpreted as a signature of amplitude noise.

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5.3 Phase-locking For the phase locking of the THz mode corresponding to the lowest frequency line (44MHz) in Fig.15 to a harmonic of the repetition rate of the fs-laser we exploited the technique detailed in [23]. A schematic of the phase-locking circuit is shown Fig.17. The downconverted THz line, at frequency f, is mixed with the output of an RF generator, at fRF = 44MHz to produce an error signal (f - fRF). The error signal is finally fed into a home-made phase-lock electronics that allows controlling the QCL current within a bandwidth of a few MHz. At the same time (not shown) the QCL is RF modulated at 14GHz with a second RF generator. A third RF synthesizer (not shown in the Fig.) is used to lock the fs-laser repetition rate using a piezoelectric transducer to control the laser cavity length. The three RF generators share a common 10MHz clock.

Fig. 17. Schematic of the setup used to phase-lock the 44MHz line (blue arrow) of Fig. 15 to a harmonic of the repetition rate of the fs-laser (see text).

Acknowledgments

We are grateful to two anonymous reviewers for their helpful comments and to Lorenzo Columbo for helpful discussions on multi-mode dynamics. We also acknowledge partial financial support from the European Community research project TERACOMB (call identifier FP7-ICT-2011-C, grant no. 296500).

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