Self-starting Harmonic Comb Emission in THz Quantum Cascade Lasers
Andres Forrer, Yongrui Wang, Mattias Beck, Alexey Belyanin, Jérôme Faist, Giacomo Scalari
MManuscript Title
Self-starting Harmonic Comb Emission in THz Quantum Cascade Lasers
Andres Forrer, a) Yongrui Wang, Mattias Beck, Alexey Belyanin, Jérôme Faist, and Giacomo Scalari b) Institute for Quantum Electronics, Department of Physics, ETH Zürich, 8093 Zürich,Switzerland Department of Physics and Astronomy, Texas A&M University, College Station, TX,77843 USA
Harmonic comb state has proven to be emerging in quantum cascade lasers and promoted by an interplay betweenparametric gain and spatial hole burning. We report here on robust, pure, self-starting harmonic mode locking inCopper-based double-metal THz quantum cascade lasers. Different harmonic orders can be excited in the same lasercavity depending on the pumping condition and stable harmonic combs spanning more than 600 GHz bandwidth at 80K are reported. Such devices can be RF injected and the free running coherence is assessed by means of self-mixingtechnique performed at 50 GHz. A theoretical model based on Maxwell-Bloch equations including an asymmetry inthe gain profile is used to interpret the data.Quantum Cascade Lasers (QCLs) are compact and pow-erful semiconductor frequency comb sources in the mid-IR and THz regime exhibiting frequency modulated (FM) out-put with a linear chirp over the round trip time . They havebeen investigated in their properties, stabilization and controlin the mid-IR and partially in the THz region . Lately,harmonic frequency combs operating on multiples of the fun-damental round trip frequency have been reported and investi-gated in mid-IR Quantum Cascade Lasers (QCLs) as wellas in actively mode-locked THz QCLs . Harmonic combsemitted by mid-IR QCLs were observed only after carefulcontrol of the optical feedback . Finally, controlled gen-eration of specific harmonic comb states can be achieved byoptical seeding or by defect-engineered mid-IR QC ringlasers .In this Letter, we present results on robust, self-startingharmonic frequency combs in double-metal Au-Au and Cu-Cu THz QCLs up to temperatures of 80 K. First, we discussdifferent THz QCLs which show at multiple bias-points self-starting and pure harmonic comb states and report on theirexperimentally observed differences with respect to mid-IRQCL harmonic combs. Successively, the coherence of thespectral modes is examined via a self-mixing technique onthe free running electrically detected beatnote. We demon-strate as well injection locking of the harmonic state to an RFsynthesizer. The observed harmonic comb state with an asym-metry in the spectral emission is theoretically discussed usinga Maxwell-Bloch approach including two lower lasing states.The investigated THz quantum cascade active region is ahomogeneous, four quantum well design based on highly-diagonal transition. The upper-level has two transitions totwo lower-levels with dipole moments of z ul ∼ . z ul ∼ . a) Electronic mail: [email protected] b) Electronic mail: [email protected] region itself was previously studied in the fundamental fre-quency comb regime including injection-locking in Refs. .The active region is embedded in a double metal gold-gold(Au-Au) resp. low loss copper-copper (Cu-Cu) waveg-uides featuring lossy setbacks in the top cladding for trans-verse mode control. As was mentioned in Ref. the de-vices show spectral indications of self-starting harmonic combstates. In the case of Au-Au waveguides lasing starts in a sin-gle mode regime and evolves into the harmonic state, finallybreaking into the fundamental/dense comb state. This mech-anism mainly follows the observations on mid-IR QCLs .In the case of Cu-Cu waveguide we observe a different be-haviour: the harmonic comb does not arise gradually froma single mode but emerges spontaneously alternating withthe fundamental comb or high-phase noise states. The har-monic comb state is observed at different bias-points, evenmuch higher than the laser threshold. An example of a 4.00mm long and 64 µ m wide device exhibiting self-starting har-monic comb state from a dense state is presented in Fig. 1(a)for currents from 520 mA to 555 mA and from 780 mA to814 mA, operating at 80 K and with a lasing threshold of390 mA. The presented spectra are zero-padded, apodizedwith the Blackman-Harris window, and a phase correctingalgorithm is applied. The symmetry of the interferogram(IFG) arising from equally spaced modes and its interpretationin terms of coherence is further discussed below and in Fig. 2.In Fig. 1(b) we present the intensity spectrum of a fundamen-tal comb at 755 mA bias current. The electrically detectedbeatnote at 9.91 GHz is detected over a bias-tee with 67 GHzspectrum analyzer (SA, Rohd&Schwarz, FSU67) and the RFspectrum is shown in Fig. 1(c). The observed linewidth is inthe sub-kHz range. The harmonics of the fundamental beat-note resp. the beating of wider spaced modes are visible aswell. We observe up to the 6th harmonic (60 GHz) of thefundamental indicating coherence at least up to the 6th mode.The observed rise of the noise floor at 50 GHz is an intrin-sic effect of the spectrum analyzer (hashed area). The insetshows the fundamental beatnote. Fig. 1 d), measured on thesame device, shows the harmonic comb state skipping fourmodes at 800 mA driving current. Additionally to the ob-served THz harmonic spectrum, the intra-cavity mixing of thelasing modes spaced by 5 roundtrips leads to a current modu-lation on the bias-line which is as well detected over the bias- a r X i v : . [ phy s i c s . op ti c s ] F e b anuscript Title 2 I n t e n s i t y S p e c t r u m ( d B / d i v ) I n t e n s i t y S p e c t r u m ( d B / d i v ) f rep
20 40 60Frequency (GHz)11010090807060 R F P o w e r ( d B m ) R F P o w e r ( d B m ) instrumental featureinstrumental feature d B m
60 80 100Frequency (kHz)+49.346 GHz 120110100908070 d B m I n t e n s i t y S p e c t r u m ( d B / d i v )
520 mA538 mA550 mA555 mA794 mA
800 mA
814 mA780 mA a) b)c) d)e) x f rep FIG. 1: Harmonic comb state in a 4 mm long and 64 µ m wide double-metal Cu-Cu THz QCL at 80 K. a) Transition from adense state to a harmonic comb state and back for bias current from 520 mA to 555 mA and from a dense state to a harmonicstate from 780 mA to 814 mA close to roll-over. Lasing threshold is at 390 mA. b) Intensity spectrum with phase-correction,apodization and zero-padding applied of a fundamental comb state. c) Electrically detected RF spectrum on the biascorresponding to the fundamental comb sate with a narrow beatnote at 9.91 GHz and its harmonics. The increasing noise floorand the transition at 50 GHz is an instrumental feature of the SA used. The inset shows a zoom on the fundamental BN. d)Intensity spectrum of a harmonic comb state with phase-correction, apodization and zero-padding applied. The modes arespaced by 5 × FSR of the cavity. e) Electrically detected single beatnote at 49.346 GHz on the laser bias. The absence of otherbeating signals indicates the purity of the harmonic state. Inset: Zoom on the harmonic beatnote with a RBW of 300 Hz and alinewidth of 850 Hz with optical feedback from the FTIR.tee. The single beatnote at 49.346 GHz is shown in Fig. 1(e)corresponding to the expected 5th harmonic of the cavity rep-etition rate (49 . / = . ∼
850 Hz and a ∼ .Compared to their Au-Au waveguide counterparts, the Cu-Cu waveguide features lower losses (4.4 cm − resp. 5.7 cm − at 3 THz for 150 K lattice temperature, 2D Comsol® 5.6 , mir-ror losses and intersubband absorption excluded.) which leadto increased intracavity fields (roughly 0.2 kV/cm at peak) andtherefore increases the effects of parametric gain, discussedin the theory section below. This would explain the morefrequently observed self-starting harmonic states at differentfractions of the threshold current in Cu-Cu devices, which donot only arise from a single-mode instability close to thresh-old. Multiple of these observed harmonic states appear fordifferent cavity lengths and widths as well as different temper-atures. Fig.2 a) shows a series of IFGs with the correspondingspectra in Fig. 2 b). The center of the envelope of the modes,in contrast to mid-IR cases, is found to be in-between two modes. This asymmetry in the envelope marks a differencein respect to what was observed in mid-IR QCL where a cen-tral, more intense mode and symmetric, less intense harmonicmodes are frequently observed .As mentioned above, the coherence of any comb or har-monic state can be verified to a certain degree by compar-ing the symmetry and periodicity of the measured IFG. Theequidistant spacing of the modes in a fundamental or har-monic comb leads to a periodicity of the beating signal, rep-resented by the delay, in the IFG. This means that for a per-fect interferometer all modes are in-phase for delays corre-sponding to a multiple of the mode spacing ( dx = c / f rep ) andthe measured IFG should identically repeat itself. For non-equidistant modes the retardation for being in-phase is dif-ferent for different mode pairs. Therefore, mode pair beat-ings are getting out-of-phase for increasing delay. By nu-merically generating IFGs similar to our case but with non-equidistant spacing we find that a linear increase of ∼
50 MHzin the mode spacing still leads to a visible asymmetry, see alsoSupplement, section VI. This 50 MHz is much smaller thanthe nominal Fourier Transform resolution of 2.5 GHz of ourFTIR. In the case of a comb with more modes this symmetryargument is even valid down to the MHz level. The IFGs inFig. 2(a) exhibit such a symmetry and a periodicity except thelast one. It is important to notice that the envelope is slowlydecreasing due to diffraction losses and beam divergence ofany real FTIR measurement and its alignment but the symme-anuscript Title 3try within each period is conserved. The last IFG and spectrain dark blue shows an example of a harmonic-like state that isnot pure. This is shown in Fig. 2(c) where a zoomed versionof the third IFG from top of the harmonic state (light blue)and the IFG of the harmonic-like state (dark blue) around theZPD and close the the maximum travel range of the FTIRare presented. The symmetry is preserved for the harmoniccomb whereas for the harmonic-like state a clear asymmetryis observed. This IFG symmetry argument helps to identifynon-pure comb states even in the presence of a single visibleelectrical beatnote or strong injection. Of course the symme-try in the IFG is a required property for a comb (fundamentalor harmonic) and can quantify to a certain extent the coher-ence between the modes. The coherence should be furthertested by SWIFT , Intermode Beatnote Spectroscopy , DualComb or any suitable coherence measurement . Most ofthese techniques require fast detectors, but a much simplerapproach based on the self-mixing, where the QCL itself actsas a fast heterodyne detector , can verify to some extent thecoherence of the modes and was first presented in Ref. . J / J f rep J / J f rep J / J f rep S p e c t r a ( a . u . ) I n t e n s i t y S p e c t r u m ( d B / d i v ) J / J
15 x f rep J / J
10 x f rep J / J
10 x f rep J / J f rep a) b)c) FIG. 2: a) shows a series of IFG of harmonic states (lightblue) and the harmonic-like state (dark-blue). b) presents thecorresponding spectra which all show similarities toharmonic comb states. c) compares the symmetry argument,which is a required property of the harmonic comb (as wellfor fundamental combs), from the harmonic comb state (lightblue) and the harmonic-like state (dark blue), where in thelater the symmetry is not preserved.The self-mixing coherence setup is sketched in Fig. 3(a)where a 4 mm long and 64 µ m wide QCL is aligned to aFTIR and the electrical beatnote is detected over the bias-line. It is important to notice that THz devices are intrinsi- cally less sensitive to feedback as compared to mid-IR lasers,due to the high impedance mismatch of the double metalwaveguide that provides high facet reflectivity. This allowsthe use of self-mixing techniques that are not destroying thecomb state as observed in the mid-IR . The FTIR operates instep-scan where for each step the beatnote intensity and fre-quency is recorded. From the single mode self-mixing theoryof Lang-Kobayashi a weak feedback of the optical mode it-self will lead to a slight frequency shift of the optical mode.In the experiment the feedback comes from the FTIR that fil-ters the optical modes as well. In the case of a frequencycomb each mode experiences a feedback at its own frequencywhich will lead to its shift. Since all modes are locked co-herently the shift induced to one mode will influence all othermodes and their spacing and therefore the beatnote frequencyhas to be adjusted slightly. This allows one to measure theeffect of self-mixing by the frequency change of the beat-note of a laser in a comb state. A more rigorous mathemat-ical discussion of this concept was first presented in Ref. .We already applied this self-mixing intermode beatnote spec-troscopy (SMIB) technique on a single double-metal (Au-Au)THz waveguide emitting simultaneously two unlocked combsspaced by an octave with two independent beatnotes . Weshowed that the detected self-mixing signal is sensitive to itsgenerating comb but not to the other, indicating that indeedonly coherent modes determine the beatnote shift whereas in-coherent do not. This approach is applied to the harmonicstate presented in Fig. 1 and the uncorrected SMIB IFG isshown in Fig. 3(b) and the spectrum in (c). The induced beat-note change is on the order of 10 − of its frequency and theslow drift arise from slight temperature changes. By compar-ing the SMIB spectra with the DC FTIR spectra we see thatall modes are coherently locked and produce the beating sig-nal showing the coherent harmonic comb state. It has to benoted that the SMIB can not fully verify the degree of coher-ence as it does not provide any information about the relativephases of the modes.RF injection locking is widely used for comb repetition ratestabilization . In the following a strong free running nar-row beatnote, as shown in Fig. 4(a), is injection locked aswell. The RF signal is set to a frequency roughly 600 kHzaway from the beatnote. While increasing the injection powerfrom -18 dBm at the synthesizer output up to 5 dBm typicalpulling, appearance of sidemodes and final locking at 2 dBmis observed as presented in Fig. 4(b). The locking range at2 dBm is roughly 1.2 MHz.Besides the experimental findings of harmonic combs inTHz QCLs and their stabilization and coherence measure-ment, we develop a theoretical model supporting their forma-tion and appearance. We argue that the harmonic comb regimein our lasers can be explained by the interplay of two opti-cal transitions in the active region. This is in fact a commonfeature of THz QCLs with a diagonal transition design. Thepresence of two optical transitions with different but compa-rable dipole moments favors the proliferation of lasing modesseparated by several FSRs. To be specific, here we use anactive region model which includes one common upper laserstate and two lower laser states, (see Fig.1s of Supplement).anuscript Title 4 B N F r e q . ( k H z ) + . G H z I n t e n s i t y ( a . u . ) Self-Mixing Interbeatnote Spec.DTGS rapid scan
QCL FTIRSACurrentSource a)b)c)
FIG. 3: Self-mixing intermode beatnote (SMIB)spectroscopy experiment. (a) sketches the setup where theFTIR provides a slight feedback as a function of time and theinduced beatnote frequency shift is recorded by the SA. (b)shows the uncorrected SMIB IFG. The slow drift arises fromtemperature instabilities. (c) compares the SMIB spectrumwith the FTIR spectrum showing the coherence of the modes.Following the approach similar to the one in , we calculatethe gain, amplitudes, and phases of weak sidebands generatedin the presence of a single strong lasing mode. The details ofthe calculations and the numerical parameters are describedin the Supplement. The formalism is based on the density-matrix equations coupled to the wave equation for the EMfield. Since the frequencies of the optical transitions are closeto each other, around 11 and 14 meV according to bandstruc-ture calculations, both of them contribute to the optical po-larization and laser field. First, we find the field of a stronglaser mode in the third-order nonlinear approximation; thenwe calculate the gain and eigenvalues of weak side modes.At least two side modes have to be included as they are cou-pled through a strong central mode in the four-wave mixingprocess. Figure 5 presents the net gain, the amplitudes, andthe phases of weak side modes for three different cases: (i)only one optical transition with the dipole moment of 3.7 nm;(ii) two symmetric optical transitions with equal dipole mo-ments equal to 2.6 nm each, and (iii) two asymmetric opti-cal transitions with dipole moments equal to 3.0 nm and 2.1nm. The dipole matrix elements in the first two cases are cho-sen so that the values of d ul α + d ul β are unchanged from thethird case, and hence the total gain coefficients in all casesare similar. From Comsol simulations of waveguide modes,we took the total cavity decay rate corresponding to the fieldpropagation loss of 6.5 cm − and group velocity dispersion
750 500 250 0 250 500 750Frequency (kHz)+ 49.382222 GHz offset P o w e r ( d B m ) free runningRBW = 10 kHz1.5 1.0 0.5 0.0 0.5 1.0 1.5BN Frequency (MHz)+ 49.342365 GHz offset P o w e r ( d B / d i v ) + o ff s e t -18 dBm-14 dBm-11 dBm-7 dBm-2 dBm-1 dBm0 dBm1 dBm2 dBm3 dBm4 dBm5 dBm a)b) FIG. 4: (a): free-running beatnote for the laser operating onthe 5th harmonic state. (b): beatnote injection locking as afunction of increasing RF power power (from the bottom).Sidebands are also visible and dashed lines are guides to theeye.(GVD) of 6 . × fs /mm. Comparing the gain spectra inFig. 5(a) one can see that for a single optical transition or twosymmetric optical transitions the net gain of weak side modesis positive starting from zero detuning, which indicates thatmultimode lasing can start from adjacent modes favoring thedense laser spectrum. Only in the case of two asymmetric op-tical transitions the net gain of side modes is negative at smallfrequency detunings, preventing lasing on adjacent modes andfavoring the harmonic state.Figure 5(b) shows that the net gain of the weak side modesfor two asymmetric optical transitions increases with pump-ing level at relative large frequency detunings. At the sametime, the suppression of gain around zero frequency detun-ing is stronger at higher pumping levels. This feature doesnot persist when pumping level is further increased (result notshown), which indicates that harmonic state is more favoredat a specific range of pumping levels.Fig. 5(c) shows that in the case of two asymmetric opticaltransitions the phase difference between the two side modes isaround π . However, the amplitudes are very different, whichindicates that the laser field is not FM, it will have strongamplitude modulation. In contrast, in the case of one opti-cal transition and two symmetric optical transitions, the twoweak side modes have similar amplitudes at small frequencydetunings ( <
30 GHz), and their phase relations indicate FMfield. At larger frequency detunings ( >
50 GHz), the two sidemodes have different intensities, which is due to the effect ofGVD. See the Supplemental for the details on these two cases.anuscript Title 5 -200 0 200
Frequency detuning (GHz) -2-1012 N e t ga i n ( s - ) (a) -200 0 200 Frequency detuning (GHz) -2-1012 N e t ga i n ( s - ) (b) p = 1.10p = 1.05p = 1.01 (c) FIG. 5: The instability of weak side modes and their intensities and phases in the presence of a central strong lasing mode as afunction of frequency detuning from the central mode for the active region. (a) The net gain at pumping level of p = . p = 1.01, 1.05, and1.1, from bottom to top at frequencies larger than 100 GHz. (c) The intensities and phase difference of the weak side modes atpumping level of p = 1.1. The gain relaxation times are T = 20 ps and T = 0.2 ps, and the population grating diffusion factor is D = cm / s . Horizontal lines in (a) and (b) indicate zero net gain.The results of the linear analysis of the multimode genera-tion for two asymmetric optical transitions are qualitativelyconsistent with the harmonic lasing state with asymmetricsidebands observed in the experiment. The limitations ofweak-sidemode approximation prevent us from making anyquantitative predictions. The actual multimode lasing statewill be determined by the multiwave mixing of many stronglaser laser modes. Its modeling requires fully nonlinear space-time domain simulations which is beyond the scope of thepresent paper. Still, the presented analysis allows us to fol-low analytically how the asymmetric harmonic state emergesfrom the coherent interplay of two optical transitions, whichcould be a physical mechanism behind the self-starting har-monic comb emission in THz QCLs. Obviously, more studiesare needed before a complete physical picture is revealed. It isfascinating, however, how QCLs continue bringing new sur-prising features to such a well studied field as fundamentallaser dynamics. ACKNOWLEDGMENTS
We acknowledge the financial support from H2020 Euro-pean Research Council Consolidator Grant (724344) (CHIC)and from Schweizerischer Nationalfonds zur Förderung derWissenschaftlichen Forschung (200020-165639). Y.W. andA.B. acknowledge the support from NSF grant No. 1807336.We thank the group of U. Keller for lending the 50 GHz signalgenerator and thank Martin Franckié for the discussion.
DATA AVAILABILITY
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