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Quantum Cascade Laser Frequency Combs

Jerome Faist,1 Gustavo Villares,1 Giacomo Scalari,1 Markus

Rosch,1 Christopher Bonzon,1 Andreas Hugi,2 and Mattias Beck1

1Institute for Quantum Electronics, ETH Zurich, Switzerland, E-mail: jfaist@ethz.ch2IRsweep GmbH, Zurich, Switzerland, E-mail: andreas.hugi@irsweep.com

(Dated: November 2, 2015)

It was recently demonstrated that broadband quantum cascade lasers can operate as frequencycombs. As such, they operate under direct electrical pumping at both mid-infrared and THz fre-quencies, making them very attractive for dual-comb spectroscopy. Performance levels are contin-uously improving, with average powers over 100 mW and frequency coverage of 100 cm−1 in themid-infrared. In the THz range, 10 mW of average power and 600 GHz of frequency coverage arereported. As a result of the very short upper state lifetime of the gain medium, the mode prolifer-ation in these sources arises from four wave mixing rather than saturable absorption. As a result,their optical output is characterized by the tendency of small intensity modulation of the outputpower, and the relative phases of the modes to be similar to the ones of a frequency modulated laser.Recent results include the proof of comb operation down to a metrological level, the observation of aSchawlow-Townes broadened linewidth, as well as the first dual-comb spectroscopy measurements.The capability of the structure to integrate monolithically non-linear optical element as well as tooperate as a detector show great promise for future chip integration of dual-comb systems.

I. INTRODUCTION

An optical frequency comb1 is an optical source witha spectrum constituted by a set of modes which are per-fectly equally spaced and have a well-defined phase re-lationship between each other. Frequency combs can beobtained by exploiting the natural phase locking mecha-nism arising when lasers operate in mode-locking regimeproducing ultrafast pulses. As a result, the ensemble ofcomb frequency lines at frequencies fn, given by

fn = fceo + nfrep (1)

are spaced by the repetition rate of the laser frep whichcan be made extremely stable and locked to an externalmicrowave source. In addition, the ensemble of frequencylines can be shifted by the so-called carrier-envelope off-set frequency fceo

1. As illustrated in Fig. 1, the differencebetween an array of single frequency lasers and an opti-cal comb is the correlation in the noise of each individ-ual line, immediately apparent when considering noiseterms added to fceo and to frep. Whereas the hetero-dyne beat between two independent single mode opticalsources with similar linewidth δfn will yield a signal witha linewidth δνRF =

√2δfn, the same experiment per-

formed on a comb will yield a value that may be muchbelow the one of the individual lines because of the cor-relation between the noise of the lines. This very pecu-liar relationship between the modes enables the conceptof self-referencing. In a mode-locked laser broadened tomore than an octave1, by beating the second harmonic ofa line in the red portion with a line in the blue portion ofthe spectrum, the offset frequency fceo of the comb can bedirectly retrieved and stabilized2. As a result, the abso-lute optical frequency of each comb line is rigidly linke-Cundiff:2003umd to the microwave reference frequencyfrep, allowing optical frequency combs to act as rulers inthe frequency domain. By enabling extremely accurate

Array of N independent DFBs combined:

line-to-line frequency noise is uncorrelated

Optical frequency comb:

line-to-line frequency noise is correlated

Noise

Locking

ω

ω

FIG. 1: Schematic difference between an array of single modelasers (top) and a frequency comb (bottom).

frequency comparisons using a direct link between themicrowave and optical spectral ranges, frequency combshave opened new avenues in a number of fields, includingfundamental time metrology1,3, spectroscopy as well asfrequency synthesis. In addition, they also had a tremen-dous impact on many other fields such as astronomy,molecular sensing, range finding, optical sampling, andlow phase noise microwave generation4. Their fundamen-tal significance and impact on science was reflected bythe attribution of the Nobel Prize in Physics in 2005 toTheodor W. Hansch and John L. Hall5.

Besides mode-locked lasers, optical frequency combshave recently also been generated using high-Q micro-cavity resonators pumped by narrow linewidth continu-

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ous wave lasers6,7. In this case the Kerr non-linearity isresponsible for establishing the stable phase relationshipbetween the laser modes. In contrast to mode-lockedlasers, Kerr combs can exhibit complex phase relationsbetween modes that do not correspond to the emission ofsingle pulses while remaining highly coherent8. Opera-tion of a Kerr comb in a pulsed regime with controlled for-mation of temporal solitons was recently demonstrated9.

An extremely appealing application of optical fre-quency combs is the so-called dual-comb spectroscopy,where multi-heterodyne detection is performed allowingFourier transform spectroscopy with potentially high res-olution, high sensitivity and no moving parts10,11. Asshown schematically in Fig. 2, two combs with slightlydifferent repetition rates are used in a local oscillator-source configuration. Indeed, spectroscopy by meansof optical frequency combs surpassing the precision andspeed of traditional Fourier spectrometers by several or-ders of magnitude was recently demonstrated12–14. The

FIG. 2: Principle of dual-comb spectroscopy. a Schematicdiagram of a dual-comb spectroscopy setup. b Schematic di-agram in the frequency domain.

use of dual-comb spectroscopy is especially interesting inthe mid-infrared portion of the spectrum, the so-called“molecular fingerprint region” where most fundamentalroto-vibrational absorption bands of light molecules canbe found. Applications of mid-infrared spectroscopy arein the areas of environmental sensing, including isotopo-logues, medical, pharmaceutical, and toxicological mea-surements as well as homeland security applications formolecules that are related to explosives. The THz re-gion is also of high importance for non-invasive imaging,astronomy, security and medical applications15,16. Forthese reasons, there is a very strong demand to also cre-ate optical frequency combs centered in the mid-infraredand THz regions of the spectrum17. A possible approachis to downconvert the near-infrared emission by means

of non-linear optics, still maintaining their short-pulsenature18. Direct mid-infrared emission, down to 6.2µmwavelength, has been also achieved using direct pumpingof an OPO using a Tm-doped fiber laser19. Also very in-teresting are the results achieved using high-Q microres-onators based on Si waveguides20,21 or microcrystallineresonators22. Recently, such a comb was generated in amicroresonator using optical pumping from a quantumcascade laser (QCL) at λ ≈ 4.5µm23,24. Nevertheless,one common drawback of these optical sources is thatthey consist of different optical elements that must beassembled together.

This paper reviews an emerging new optical frequencycomb technology based on QCLs. Section 2 reviews thebasic physics, as well as the main operation characteris-tics of QCL combs. In section 3, the techniques devel-oped to characterize the comb operation are described.The topic of dispersion compensation is addressed in sec-tion 4. The application in spectroscopy of dual-combssystems based on QCL combs is discussed in section 5.Conclusions and outlook are reported in section 6.

II. QCL COMBS

QCL broadband technology QCLs25 are semiconduc-tor injection lasers emitting throughout the mid-infrared(3-24µm) and THz (50-250µm)26,27 regions of the elec-tromagnetic spectrum. First demonstrated in 199428 atλ ≈ 4.3µm, they have undergone a tremendous devel-opment. Their capability to operate in a very wide fre-quency range makes them very convenient devices for op-tical sensing applications. In particular, single frequencyemitter devices have demonstrated both very large con-tinuous optical output power up to 2.4 W29, high tem-perature operation in continuous wave30 and low elec-trical dissipation below a Watt31. The physics of quan-tum cascade laser is, in addition, very beneficial for theiroperation as broadband gain medium. First, intersub-band transitions are transparent on either side of theirtransition energy. Interband transitions, in contrast, aretransparent only on the low-frequency side of their gainspectrum and highly absorbing on the high-frequencyside. Second, the cascading principle almost comes nat-urally because of the unipolar nature of the laser. Thesetwo features enable the cascading of dissimilar active re-gion designs emitting at different wavelengths to createa broadband emitter. This concept, first demonstratedat cryogenic temperature32 was further developed for ap-plications with high performance, inherently broad gainspectra designs where a single upper state exhibits allowstransitions to many lower states33. This technology wasthe base to the fabrication of external cavity quantumcascade lasers with very large tunability34.Mode-locking of QCLs It was suggested very early

that broadband QCL could be mode-locked to provide ul-trashort mid-infrared pulses32. In a mode-locked laser38,the fixed phase relationship between the optical modes –

3

FIG. 3: Mode-locking QCLs. a Schematic diagram of an actively mode-locked QCL: the modulated section opens a windowin time, allowing a single pulse to propagate in the cavity. b The computed transfer function of a QCL, showing the highmodulation capability at frequencies corresponding to the round-trip frequency of a 3-6 mm cavity laser. Results publishedin25. c Active region based on a photon-assisted tunneling transition, with very long upper state lifetime. Results publishedin35. d Spectrum of the device presented in c under modulation. e Interferometric autocorrelation technique: the optical pulsesare characterized by Fourier Transform Interferometer followed by a two-photon quantum well photodetector. f The resultinginterferogram, measured 10% above threshold, shows the ratio of 8:1 between the center and the wings, expected for singlepulses. Results published in35. g Electro-optic sampling, using a femto laser comb of the output of an actively mode-lockedTHz QCL. Results published in36. h THz reconstructed electric field trace. Results published in37. i Intensity of the pulses.Results published in37.

needed for the formation of an optical comb spectrum asdescribed by Eq. 1 – is obtained by having a single pulsepropagating in the laser cavity. This mode of operationof the laser is generally obtained either by a subtile com-pensation between a negative dispersion in the cavity andthe Kerr effect, giving rise to the propagation of a tem-poral soliton, or by a saturable absorber opening a time-window of low loss for a pulsed output. The short pulsesthat will be produced at the roundtrip frequency of thecavity will have, in well designed laser, an average powercommensurate to the one of a continuous wave laser. Thisis possible because the energy of the two-level system,stored in the inverted medium, is abruptly released in theoptical pulse. In contrast, QCLs, being based on inter-subband transitions in quantum wells, exhibit very shortupper state lifetime with τ2 ≈ 1 ps39. In fact, in highperformance devices operating at room temperature, thistime is in the sub-picosecond range (τ2 ≈ 0.6 ps). Thistime is much shorter than the typical cavity round triptime τrt (64 ps for a 3 mm long device), such that the

product ωτ2 << 1, where ω = 2πfrep = 2π/τrt is the an-gular frequency corresponding to the longitudinal modespacing. As a result, passive mode-locking in the sensedescribed above is not possible.

The very short upper state lifetime is responsible forthe very fast dynamics of the transport in QCLs. For thisreason, QCLs can be modulated at frequencies above 20GHz, equal to the round-trip frequency of 3 millimeterlong cavities, as shown in Fig. 3b. Active mode-lockingis then feasible by modulating one short section of theQCL active region at its round trip frequency (Fig. 3a).However, to mitigate the gain saturation in the active re-gion, it is necessary to work with upper state lifetimes inthe tens of picoseconds, such that the condition ωτ2 ≈ 1is fulfilled. Based on this principle, active mode-lockinghas been achieved in the mid-infrared with active regionsbased on photon-assisted tunnelling transitions (as shownin Fig. 3c)35 where very long intersubband lifetimes inthe order of 20 ps can be achieved40. At a temperatureof 77K, pulse length of 3 ps with a bandwidth of 15 cm−1

4

were achieved35 (see Fig. 3c, d and e). Nevertheless, thegain recovery time of 50 ps still restricted the operationof the device to a driving current range not much higherthan 10% above threshold, as predicted by numerical sim-ulations41.

In the THz, relatively long upper state lifetimes areachieved at cryogenic temperatures with state-of-the artactive regions based on bound-to-continuum transitions.Active mode-locking by RF injection was achieved at3 THz with pulses lengths of about 10 ps and a bandwidthof 100 GHz36,37,42, as shown in Fig. 3i. One very inter-esting aspect of these experiments was the possibility oflocking the THz QCL to a near-infrared fiber comb anduse the latter to perform electro-optic sampling of theTHz pulse (see Fig. 3g and h), as reviewed in43. Finally,mode-locking was also achieved in the THz by a combi-nation of injection seeding and RF gain switching44,45.In the latter case, in contrast to the conventional mode-locking by RF injection, the phase of the THz pulse isfixed by seeding the THz QCL laser with the output ofa gated photoconductive antenna.

However, compared to passive mode-locking in whichthe pulse length can routinely reach the inverse of thegain bandwidth Ωg, fundamental active mode-lockingleads to much longer pulses since the pulse length τ isnow a geometrical average of the gain bandwidth andthe round-trip frequency ω38. In addition, as mentionedabove, the output power is limited by the necessity toremain close to laser threshold. All in all, the reducedresulting mode-locking bandwidth and the low opticalpower therefore limits the usefulness of such devices forbroadband spectroscopy.

Further work in mode-locking QCLs include the useof a ring external cavity. This arrangement simplifiesthe mode selection because it minimizes the spatial holeburning effect, and it also allows the use of modulationfrequencies low enough to enable complete switching ofthe laser on and off. This geometry has been analysedboth theoretically46 and experimentally47,48.

Finally, an intriguing proposal is the use of so-calledself-induced transparency mode-locking, where an ab-sorber section is precisely tailored to undergo a half-Rabioscillation under the optical pulse, stabilizing the opticaloutput49. Although the concept is very general, it wasargued that the QCL, due to its short upper state life-time, is a specially well suited system to implement suchmode-locking scheme. The mode-locking was predictedto remain stable even under small amount of dispersionand saturable nonlinearity50. Nevertheless, the active re-gion design requires a very fine tuning of the respectiveoscillator strengths of the gain and absorber sections, andso far no experimental evidence of such mode-locking hasbeen reported.

Four wave mixing in QCLs The active region of theQCL exhibits optical non-linearities, as any two-level sys-tem. In particular, the very short upper state lifetime,that strongly hinders fundamental mode-locking, is re-sponsible for a very broadband four wave mixing pro-

cess. Four wave mixing due to intersubband transitionwas first measured in a doped multiquantum well51. Ina QCL active region under operation, measurements ofthe four wave mixing (FWM) product was performed byinjecting two single frequency sources and analysing theoutput spectrum52. As shown schematically in Fig. 4a,a single mode DFB QCL and the output of a tunablesource based on difference frequency generation were in-jected in an anti-reflection coated QCL amplifier, drivenin continuous wave at room temperature close to its max-imum current. The QCL amplifier consisted of a singlestack of strain-compensated InGaAs/AlInAs active re-gion operating at 4.3µm wavelength53. For these oper-ation conditions, the computed upper state lifetime isT1 = 0.29 ps and dephasing time is T2 = 0.14 ps. Asshown in Fig. 4 b, the output showed the expected mix-ing product at frequencies corresponding to ω4 = 2ω1−ω2

and ω3 = 2ω2−ω1. As shown in Fig. 4c, the dependenceof FWM signals as a function of detuning was in goodagreement with the value of the effective nonlinear sus-ceptibility χ(3) computed using a two-level density matrixmodel and given by:

χ(3)(∆, δω) =2e4∆Nz4ij

3ε0~3(δω −∆− i/T2)

(∆− δω + i/T2)

× (−δω + 2i/T2)(∆ + i/T2)−1

(δω − i/T1)(δω −∆− iT2)(δω + ∆− i/T2)

(2)

where ∆ = ω1 − ω12 is the detuning between the pumpand the intersubband transition, δω = ω2 − ω1 the sep-aration between the pump and probe, zij = 1.6 nm thedipole matrix element and ∆N the population inversionat threshold. For a detuning ∆ of 210 GHz, an effec-tive χ(3) = 2.5 × 10−15 m2V−2 is predicted, close to themeasured value of χ(3) = (0.9±0.2)×10−15 m2V−2. Theimportant conclusion of these measurements was that theQCL active region presents a large, resonant χ(3), aris-ing from the active medium itself. In addition, the band-width of the FWM process is much wider than the onein interband semiconductor lasers because of the muchshorter upper state lifetime.Waveguide dispersion The ability of four wave mixing

to generate mode proliferation - and ultimatly comb oper-ation - depends critically on the group velocity dispersionof the cavity. If one considers a single active region witha gain bandwidth of 100 cm−1, the latter corresponds toabout Nm = 200 cavity modes for a 3 mm long device.The finesse F of a typical QCL Fabry-Perot cavity is verylimited, but even assuming a value of F = 5 requires thatthe fractional change of the group effective index is notlarger than δ(ω)/ω = δng/ng = 1/(NmF) = 10−3 overthe gain bandwidth. The spectral change of the grouprefractive index is quantified by the group velocity dis-persion (GVD), defined as

GVD =∂

∂ω

1

vg=

1

c

∂ng∂ω

. (3)

Using the numerical values reported above, the GVDmust remain below GVD < 560 fs2/mm for the modes

5

0.1 2 3 4 5 6 7 8 9 1 2 3 4 5

Measured FWM Modeled FWM

Difference frequency ω/2π (THz)

FWM

sign

al (a

.u.)

10

1

0.1

0.01

0.200.150.100.050.00

Det

ecto

r Sig

nal (

V)

70.570.069.569.068.5Frequency ω/2π (THz)

DFB-QCL x 0.01 ω2

DFGx 0.01 ω1

FWM-upω3

FWM-downω4

DFB-QCL

DFG

QCL amplifier

ω2+/−Δω

Spectrometer

a)

b)

c)

ω1

ω2

FIG. 4: Four wave mixing in a QCL active region. aSchematic diagram of the experimental setup. b Measuredspectrum for a source separation δω = ω2 − ω1 of 314 GHz.c response of the FWM as a function of detuning betweenthe two sources (points), compared with the predictions of atwo-level model (dashed lines). Results published in52.

to remain efficiently coupled by four wave mixing. Thisnumber is of course only a rough estimate yielding theorder of magnitude of the dispersion that is relevant forcomb operation.

The GVD in an active QCL consists of three maincomponents: the dispersion of the material, the modaldispersion of the waveguide (including both lateral andvertical confinement) and the gain medium itself. Wehave then

GVD = GVDmat + GVDmod + GVDgain (4)

One very favorable feature of the mid-infrared QCL isthe fact that they operate in a transparency region farfrom both the fundamental gap as well as from the re-strahlen band. For this reason, as shown in Fig. 5a, theGVD of InP has a very low value (< 1000 fs2/mm in theregion covering the mid-infrared range from 3.3-10µm(≈ 1000 − 3000 cm−1). Similar low values are expectedfor the undoped InGaAs and AlInAs materials constitut-ing the active region. In contrast, the situation at thetelecom wavelength of 1.55µm is that InP has alreadya GVDmat ≈ 2000 fs2/mm while the narrower gap con-finement waveguide layers will have even larger values ofGVDmat.

The dispersion of the resulting ”empty” waveguide,

FIG. 5: Contributions to the waveguide dispersion. a GVDfor bulk InP material. b GVD due to the mode confinement,computed by performing a two-dimensional simulation, show-ing the impact of the ridge width on the GVD. Inset: modeprofile of a typical mid-infrared QCL.

taking into account the effect of vertical and horizontalconfinement is displayed in Fig. 5b. In that figure, thevertical waveguide geometry was the one of the originalwork reported54. Not surprisingly, strong lateral con-finement is shown to add a positive contribution to theGVD, and the choice of the final waveguide width willbe the result of a compromise between the need to keepthe total GVD low and the necessity to guarantee a sin-gle transverse mode emission. Note also that the originalreport in54 the waveguide GVD was computed combin-ing the GVD from the horizontal and vertical confine-ments separately. We found that the latter procedurewas a relatively poor approximation for tightly confinedwaveguides.

Finally, the gain itself is responsible for a group veloc-ity dispersion because of the change of refractive indexfollowing Kramers-Kronig relations. Shown in Fig. 6 isthe computation of the complete QCL, including gain,comparing the case of the three stack active region device(in solid line) with a hypothetical single stack (dashedline). The parameters are the ones of the broadband de-vice exhibiting comb operation54. At maximum power,the device is operating with a continuous spectrum cov-ering 250 cm−1, implying that the net gain is equalizedover this bandwidth. For this reason, the GVD origi-nating from the gain exhibits a wide flat minimum inthe middle of the gain curve, in contrast to the case ofthe single stack QCL. In this simulation, the total GVD

6

FIG. 6: Computation of the GVD for the active region andeffect of the gain bandwidth. Blue dashed line: a narrow-band single stack active region. Solid red line: active regionconsisting of three stack. Results published in54.

(solid black line) was predicted to exhibit a relativelywide flat minimum in the region around 1450 cm−1, ingood agreement with our observation of comb operationin the center of the gain curve.

One difficulty arising from the computation done hereis that the shape of the GVD depends strongly on theshape of the gain curve, which itself is determined self-consistently by gain saturation under operation. Thisspecial situation arises because, as compared to the caseof solid-state or fiber lasers, the value of gain in QCL islarge such that dispersion and gain cannot be separatedeasily. Treating both contributions on the same footingis one of the nice feature of the Maxwell-Bloch modelpresented in the next paragraph.

Theoretical models for comb operation The dynami-cal behavior of a multimode QCL laser is the result ofa complex interplay between the intersubband polariza-tion in the active region and the electromagnetic energystored in the optical modes of the laser cavity. The equa-tions of motion for the carriers in the active region can berepresented by the optical Bloch equations for the den-sity matrix describing the laser transition as a two-levelsystem. The Maxwell equations describe then the mo-tion of the electromagnetic wave subjected to the mat-ter polarization and losses in the cavity. The resultingMaxwell-Bloch coupled equations are the basis for theexplanation of laser dynamics.

Narrowband modulation at the round trip frequencyof the output power of continuous wave QCLs werefirst interpreted using a set of Maxwell-Bloch equationexpressed in the time domain41,55,56. However, theseequations, while well suited to describe the generationand propagation of a pulse in the laser cavity, are ill-conditioned to study the output of a laser that exhibitsmostly a frequency modulated output.

Another approach consists solving the system in thefrequency domain, using the modes of the laser cavity asa basis. This approach was already used by Lamb in 1964

FIG. 7: Schematic drawing showing the coupling between theoptical Bloch equations, describing the motion of the densitymatrix and the Maxwell equation for the optical modes.

to study the ”three-mode laser”, in which a third modeis frequency locked by the beating of two other modesthrough the four wave mixing57. The method followedby Khurgin et al.58 uses this approach and applies it toan arbitrary set of modes N as shown schematically inFig. 7. The final equation for the time evolution of the(slowly varying and normalized) complex amplitude Anof mode n is given by 58,59:

An =

Gn − 1︸ ︷︷ ︸Net gain

+i

(ω2n − ω2

nc

2ωn

)︸ ︷︷ ︸Cavity dispersion

An

−Gn∑k,l

CklBklAmAlA∗kκn,k,l,m︸ ︷︷ ︸

FWM term

. (5)

In the above equation, Gn is the transition net gain, nor-malized to the cavity losses:

Gn =1

1− inω/γ12(6)

while

Ckl =γ22

γ22 − i(l − k)ω(7)

defines the amplitude of the coherent population oscilla-tions, and

Bkl =γ122i

(1

−iγ12 − lω− 1

iγ12 − kω

)(8)

7

is the term driving the width of the four wave mixinggain. In these equations, ωnc is the angular frequency ofthe empty cavity mode, ωn = nω is the angular frequencyof one specific mode n. The broadening of the transitionis γ12 and the scattering rate of the upper state is γ22.The result of the simulation, neglecting at this stage theeffect of the dispersion, is reported in Fig. 8. The resultsshow that the short upper state lifetime prevent the for-mation of optical pulses; as a result the instantaneousintensity is almost constant (Fig. 8b) while the instanta-neous frequency swings widely over a significant portionof the output spectrum (Fig. 8d). Note that the opticalBloch equations used in the derivation of Eq. 5 are thesame as the one used for the derivation of the χ(3) inEq. 2; resulting in the expression (in normalized units)χ(3) = Gn

∑k,lBklCkl.

FIG. 8: Maxwell-Bloch model of a mid-infrared QCL comb.a Optical spectrum. b Instantaneous intensity as a functionof time. c RF spectrum of the intensity. d Instantaneousoptical frequency. Results published in58.

In insight, these results are to be expected since QCLare so-called class A laser, where the medium populationinversion and polarization can follow the optical field dy-namics. Indeed, FM mode-locking was observed in He-Negas lasers, that belong to the same class A60. In that ex-periment, however, the FM mode-locking was driven byan external phase modulator. Interestingly, it was arguedon theoretical grounds that self mode-locking throughfour wave mixing was not possible. However, it seemsthat the author did not consider the possibility of a FM-like solution61.

Figure Fig. 9 shows an experimental confirmation ofthe FM-like beahavior of a QCL comb. A sheet ofpolyethylene, which has strong wavelength-dependentabsorption (see red line in Fig. 9a), is inserted betweenthe laser output and a high-bandwidth detector. Thissheet acts as an optical discriminator that can convertthe frequency modulation of the laser output to an am-plitude modulated signal. We observe an amplification

(16 dB in this example) of the measured RF beatnote atthe round trip frequency (see Fig. 9b), as expected froma frequency-modulated signal.

FIG. 9: Frequency-modulated QCL comb. a Optical spec-trum (black) as well as polyethylen transmission spectrum(red) acting as an optical discriminator. b RF spectrum show-ing the beatnote at the round trip frequency. An increase ofthe RF beatnote, characteristics of an FM to AM conversion.

The effect of cavity dispersion and external modula-tion on the solutions of Eq. 5 was studied in ref.59. As ex-pected, a large absolute value of GVD (= 50’000 fs2/mm)prevented the formation of the comb. However, at thispoint the simulations could not conclusively assess therole of small values of GVD (≈ 500 fs2/mm) in the desta-bilisation of the coherence. Again, as expected, the samemodel could predict the formation of single pulses bymodulation of a section of the waveguide in THz deviceswith long upper state lifetime59, essentially reproducingthe experimental results observed in 36.

However, the role of electron transport is not suffi-ciently taken into account in the existing models, al-though it has been recently included in the descriptionof actively mode-locked QCLs46. In particular, some ex-perimental evidence show that embedding the QCL combin a RF microstrip waveguide improves the coherence ofthe beatnote62 while this effect is not predicted by thecomputer simulations59. A more complete theory wouldincorporate the second (slower) time scale of the injec-tion process as well as a χ(2) interaction where a modeis shifted in energy by the interaction with the RF mod-ulation at the round trip frequency.Comb operation of Mid-IR broadband QCL Fre-

quency combs are generated when the different longitu-

8

dinal modes of a laser are locked in phase1,4, creatingan array of equidistantly spaced, phase-coherent modes.In contrast with previous attempts of mode-locking ofQCLs35–37,42, broadband QCLs can achieve frequencycomb operation by using four-wave-mixing as a phase-locking mechanism54, as shown schematically in Fig. 10a.Degenerate and non-degenerate four-wave mixing pro-cesses induce a proliferation of modes over the entire laserspectrum. The dispersed Fabry-Perot modes – which arepresent in the case of a free-running multimode laserand are not phase-locked – can be injection-locked bythe modes generated by the FWM process, thus creat-ing a frequency comb. This phase-locking mechanism isvery similar to the one that occurs in microresonator-combs6,7. In addition, combined with the short gain re-covery of intersubband transitions, the laser output is notpulsed and resembles the one of a frequency modulatedlaser54. In fact, the fast gain recovery of a QCL doesnot introduce any perturbation to a frequency-modulatedlaser, as the intensity of such a laser is constant.

Fig. 10b shows a magnified view of the optical spec-trum of a QCL comb operating in the mid-infrared,at a center wavenumber of 1430 cm−1 (7µm) covering60 cm−1. We observe a single set of longitudinal modeswith no measurable dispersion within the resolution ofthe measurement (0.0026 cm−1, 78 MHz). As this mea-surement does not give any information about the rela-tive stability of the phases of the modes, different tech-niques had to be developed in order to characterize QCLcombs (see next section).

In the first place, the radio-frequency spectrum of thelaser intensity near the repetition frequency frep can bemeasured. High-bandwidth detectors capable of detect-ing radio-frequency signals up to tens of GHz are neededfor this type of characterization. Such a spectrum wouldshow a beatnote with a large linewidth (5 to 10 MHz) inthe case of two uncouples modes (such as those generatedby two independent lasers). In contrast, such a measure-ment would show a narrow beatnote in case the modesare locked in phase. The measurement of this radio-frequency spectrum, also called intermode beatnote spec-trum, is shown in Fig. 10c, for a laser biased just abovethe onset of multimode operation. It shows a beatnotewith a linewidth with a full-width at half-maximum of8.8 Hz, characteristic of comb operation.

Frequency comb operation was however not observedover the entire laser operation range in the original workof ref. 54 and these observations were also confirmedin QCL combs operating in the THz spectral range63,64.Fig. 11a shows different RF spectra together with theircorresponding optical spectra for different values of lasercurrent. In addition to the narrow RF beatnote char-acteristic of comb operation, broad beatnotes are alsoobserved, corresponding to a loss of coherence of theoptical comb. Fig. 11b shows a light intensity-current-voltage characteristics of this same device, showing therange where QCL comb operation is observed.

FIG. 10: Principle of operation of a QCL comb: a The dis-persed Fabry–Perot resonator modes are injection-locked bythe equally spaced modes generated by four-wave mixing pro-cesses. b. Magnified view of the optical spectrum of a QCLcomb operating in the mid-infrared, at a center wavenumberof 1430 cm−1 (7µm) covering 60 cm−1. c. Radio-frequencyspectrum of the intensity near the repetition frequency frep,showing a beatnote with a full-width at half-maximum of< 10 Hz. The measurement is taken at the onset of the multi-mode emission and the resolution bandwidth of the spectrumanalyser is set to 10 Hz. Results published in54.

THz QCL comb operation In addition to mid-infraredcombs, QCLs can be used to generate combs at THz fre-quencies, as the main driving mechanism is still four-wavemixing. THz QCLs can be driven in the comb regimeby careful dispersion compensation63, by engineering abroadband QCL with a flat gain curve and by a com-bination of both elements. It was recently shown thatit is possible to achieve octave-spanning lasers at THzfrequencies using QCLs64. This very wide frequency cov-erage is possible due to the gain engineering capability

9

FIG. 11: QCL comb operation regimes. a Left side: RFbeatnote spectra measured using a fast quantum well photo-conductor, together with its respective optical spectra (rightgraphs) of a QCL designed for comb operation. Above a cer-tain value of current (510 mA in this example), the RF beat-note splits into two distinctive beats. b) Corresponding lightintensity-current-voltage characteristics of the device. Theshaded area corresponds to the region where the narrow beat-note is observed. Results published in54.

typical of intersubband transitions coupled to the ex-tremely broadband nature of the double-metal waveg-uide. THz QCLs operate in double-metal cavities wherethe waveguide claddings are constituted by two metalliclayers27. In TM polarization this cavity does not presentany cutoff, making ultra-broadband operation possible.The heterogenous cascade laser used for achieving octave-spanning operation is constituted by three different ac-tive regions stacked together in the same waveguide. Itsemission spans from 1.64 THz to 3.35 THz when oper-ated in CW operation at a temperature of 25 Kelvin(Fig. 12b)64. Such a broad gain medium is beneficial forcomb operation as it results in a low and flat GVD. Fur-thermore the double metal waveguides introduce only asmall amount of GVD to the laser. As a result, theselasers can act as frequency combs similar as mid-infraredQCL combs. The characteristics of such a THz QCLcomb are displayed in Fig. 12a. The comb shown has1.55 mW of output power while spanning over a spec-tral bandwidth of 507 GHz. The beatnote exhibits alinewidth of 800 Hz limited by technical noise. So far

comb operation in these devices is limited to a spectralbandwidth of the order of 600 GHz at a central frequencyof 2.6 THz. Additionally these combs suffer from the gen-eral drawbacks of THz QCLs: operation is limited tocryogenic temperatures and feature much lower outputpower than combs in the mid-infrared (of the order of10-100 mW).Fig. 13 gives an overview over the output powers of differ-

-200 -100 0 100 200-150

-140

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nsity

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] Linewidth: 800 Hz

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1e-1

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1e-4

Inte

nsity

[a.u

.]

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.41E-4

1E-3

0.01

0.1

1N

orm

aliz

ed in

tens

ity [a

.u.]

Frequency [THz]

octave

a)

b)0.0 0.2 0.4 0.6 0.8 1.0 1.2

0

2

4

6

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10

12

14

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tage

[V]

Current [A]

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er [m

W]

0 50 100 150 200 250 300 350 400Current density [A/cm2]

com

b re

gim

e

FIG. 12: THz QCL comb. a LIV curves of a 2mm x 150µmlong laser measured at 25 Kelvin in CW operation. Theshaded area indicates the current range where the laser op-erates as a comb. The upper inset displays the intermodebeatnote, showing a linewidth of 800 Hz at a driving cur-rent of 0.9 A. The lower inset shows the corresponding opticalspectrum with a bandwidth of 507 GHz. b Octave-spanningspectrum of the same laser measured at 25 Kelvin in CW op-eration at the maximum current (1.05 A). Results publishedin64.

ent QCL-based frequency combs. The power levels rangefrom 1-10 mW for THz combs up to more than 100 mWfor mid-infrared combs. As can be seen from the samefigure, the frequency coverage of the different combs showlarge differences.

III. COMB COHERENCE

When discussing the coherence properties of a fre-quency comb, two different quantities have to be dis-tinguished. The relative coherence between the differ-ent comb modes describes the relative phase stability be-tween any two tooths of a comb. This coherence propertyis intrinsically linked to the comb repetition frequencyfrep, and a frequency comb with high degree of relative

10

FIG. 13: Average emitted power as a function of frequencycoverage for published QCL-based comb sources. Rectan-gles width indicates the frequency coverage and the averagepower refers to the integrated laser output on all the emittedwavelengths. Data from the following groups: ETHZ54,64,65,NWU66, Paris 736, PDI67, Harvard35, ENS45, MIT63.

coherence shows perfectly spaced modes. On the otherhand, the absolute coherence of a frequency comb char-acterizes the global phase stability of all modes. Thissecond coherence property is linked to the comb offsetfrequency fceo.

As QCL combs are based on a different phase-lockingmechanism as compared to frequency combs emittingpulses, new characterizations techniques had to be devel-oped to assess their coherence properties. Indeed, tradi-tional characterization techniques – such as interferomet-ric autocorrelation, FROG or others68 – are all based onnonlinear effects induced by the short optical pulse emit-ted by traditional frequency combs and cannot be usedfor the characterization of QCL combs.

For this reason, characterization techniques which didnot require pulse emission were developed. The first tech-nique developed for the characterization of the relativecoherence of QCL combs is an interferometric techniquewhere the autocorrelation of the beatnote at the combrepetition frequency is measured using a Fourier Trans-form Infrared Spectrometer (FTIR)54.

Intermode beat spectroscopy and SWIFTS The prin-ciple of intermode beat spectroscopy is schematically de-scribed in Fig. 14a. The FTIR act as an optical filterused for the spectral separation of the comb modes, andthe relative coherence is measured by investigating thebeatnote at the comb repetition as a function of the inter-ferometric delay. This is done by using a high-bandwidthdetector, sufficiently fast to measure the comb repetitionfrequency. The quantity measured is given by

IBS(ν, τ) ≡

∣∣∣∣∣F∣∣E(t) + E(t+ τ)

∣∣2∣∣∣∣∣(ν) (9)

which is the absolute value of the Fourier component ofthe intensity at the detector at frequency ν over a cho-sen resolution bandwidth of the spectrum analyser. Thequantity E denotes the time-dependent amplitude of theelectric field and F stands for the Fourier transform thatis applied to the detected intensity. The traditional in-tensity interferogram measured in a FTIR is given byIBS(0, τ). The intermode beat interferogram IBS(ν, τ) issensitive to the relative phase of the modes and can beused to measure the relative coherence of QCL combs.In analogy with Fourier-Transform spectroscopy, one cancalculate the Fourier transform of IBS(ν, τ) as

IBS(ν, ω) ≡ F(IBS(ν, τ)) (10)

where ω is usually an optical frequency. The quantityIBS(νrep, ω) evaluated at the comb repetition frequency,called the intermode beat spectrum, is shown in Fig. 14b,together with the normal intensity spectrum of the laser.One can see that both spectrum are very similar, showingthat the entire spectrum is contributing to the generationof the beatnote at the comb repetition frequency.

Another important characteristic of intermode beatspectroscopy is that one does not need to necessarilyimpose ν = νrep and can therefore measure the entireintermode beat spectrum around νrep. This can be usedto yield important information about the mechanismsthat destabilize the comb. Fig. 14c shows an example ofsuch characteristics, where the intermode beat spectrumIBS(ν, ω) was acquired over a span of 100 MHz aroundthe comp repetition frequency. In this measurement,two broad beatnotes are observed on the RF spectrum.The intermode beat spectrum IBS(ν, ω) shows that thepeak at ν = 7.4840 GHz involves mostly modes which arenear the center of the optical spectrum (ω ' 1430 cm−1)while the peak at ν = 7.4587 GHz corresponds to modeslocked for a wider bandwidth (from ω = 1360 cm−1 toω = 1500 cm−1).

Although intermode beat spectroscopy can give ac-cess to the relative coherence of the modes, it can notgive direct access to the relative phases between twomodes. For solving this issue, shifted wave interferenceFourier Transform Spectroscopy (SWIFTS) has been in-troduced63,69. A schematic description of SWIFTS isshown in Fig. 15a. The RF beatnote detected by a highbandwidth detector is now demodulated by a I/Q demod-ulator, consisting of a local oscillator (LO) creating a pairof signals different by a 90 phase shift. By phase-lockingthe comb repetition frequency to the LO, this measure-ment yields the in-phase and quadrature signals SI(τ, ω0)and SQ(τ, ω0), respectively (see Table I for a comparisonbetween the two methods). By combining both signals,the degree of relative coherence between two modes canbe obtained69

g±(ω, ω ± νrep) ≡ < E∗(ω)E(ω ± νrep) >√< |E(ω)|2 >< |E(ω ± νrep)|2 >

(11)

11

FIG. 14: Intermode beat spectroscopy. a Schematic representation of an intermode beat spectroscopy experiment. The outputof a FTIR is sent to a high-bandwidth detector, in order to measure the intermode beatnote spectrum with a spectrumanalyzer. (QWIP: quantum-well infrared photodetector. AR: anti-reflection). b Intermode beat spectrum together with thenormal intensity spectrum of the laser, showing that the entire spectrum is contributing to the generation of the beatnote atthe comb repetition frequency. c Intermode beat spectrum acquire over a large span (100 MHz), for driving conditions wherethe QCL is not operating as a comb, and yielding important informations about the comb destabilization mechanisms. Twobroad beatnotes are observed on the RF spectrum. The peak at ν = 7.4840 GHz involves mostly modes which are near thecenter of the optical spectrum (ω ' 1430 cm−1) while the peak at ν = 7.4587 GHz corresponds to modes locked for a widerbandwidth (from ω = 1360 cm−1 to ω = 1500 cm−1). Results published in54.

Fig. 15b shows the SWIFTS correlation spectrum (cor-responding to the numerator of Eq. 11) together withthe spectrum product (corresponding to the denominatorof Eq. 11). Similar to the intermode beat spectroscopy,these results analysed with SWIFTS also shows that mostof the spectral power is in the frequency comb. In ad-dition, one can also get access to the degree of relativecoherence g±(ω, ω±νrep), which is represented in Fig. 15cfor another example. This measurement shows that thedegree of relative coherence is close to unity at the tworegions of the gain spectrum where the output of the laseris maximum. However, this degree of relative coherencedecreases to zero when one get close to the edge of eachspectral region.

Finally, Table I shows a comparison between these twomethods. While both methods give access to the rela-tive coherence of the comb, intermode beat spectroscopyis sensitive to information at different frequencies thanthe comb repetition frequency and is easier to implement(no need for any phase lock loop). Complementary to in-termode beat spectroscopy, SWIFTS can give access tothe degree of relative coherence as well as to the rela-tive phases between pairs of modes, which can allow toretrieve the comb time-domain profile69. Another tech-nique has been proposed to assess the degree of coherenceof THz QCL combs. It is based on self-mixing of the laseroutput which is fed back into the laser cavity after beingfiltered by a FTIR67.Comb line equidistance using a dual-comb technique

A direct verification of the uniformity of the comb linespacing can be done by using a multi-heterodyne setup(also called dual-comb setup), as shown schematicallyin Fig. 16a70. By employing two frequency combs withslightly different repetition frequencies, this setup effec-tively maps down the frequency comb from the opticaldomain to the RF frequencies, where established count-ing techniques can be employed. The uniformity of the

mode spacing can be quantified by evaluating the devia-tion from equidistant mode spacing ε, defined as8 :

ε =fM − f0M

− fN − f0N

. (12)

where f0, fN and fM are the frequencies of threemulti-heterodyne beatnotes. After detecting the multi-heterodyne beat signal, three electrical bandpass filtersare used to isolate the beatnotes at frequencies f0, fNand fM . After amplification, three frequency countersare used to measure the frequency oscillations of the beat-notes.

In this experiment, an active stabilization is imple-mented (by acting on the current of one device) in or-der to correct the slow frequency drifts of the multi-heterodyne beat spectrum, thus preventing each beatnotefrom drifting out of the bandwidth of the electrical filters.A precise control of both trigger and gate time of all thecounters was implemented. It is particularly importantas any drift of fceo will be seen as a non-uniformity ofthe comb line spacing if the counters are not perfectlysynchronized. Even though the comb is not fully stabi-lized and frep drifts, ε should be zero at any point in timefor a perfectly synchronized setup and a perfectly spacedcomb. This method offers a direct way to measure theuniform comb spacing using counting techniques even ifthe frequency comb is not fully stabilized.

Fig. 16b shows the distribution of ε over the entire mea-surement (gate time = 10 ms, 2523 counts), showing amean value of ε = (−5.6 ± 32) mHz. Normalized to theoptical carrier frequency (42.8 THz or 7µm), this givesan accuracy of the equidistance of 7.5x10−16. This mea-surement definitely proves the equidistance of the modespacing of a QCL frequency comb, confirming the highdegree of relative coherence of QCL combs70.Schawlow-Townes linewidth of QCL combs As QCL

combs are attractive for applications such as high-

12

FIG. 15: Shifted wave interference Fourier Transform spectroscopy (SWIFTS). a Schematic representation of SWIFTS char-acterization system. The output of a FTIR is sent to a high-bandwidth detector. The detected signal is sent to an I/Qdemodulator in order to get access to the in-phase and quadrature signals, SI(τ, ω0) and SQ(τ, ω0),respectively. By combiningboth signals, the correlation of the electric field of two adjacent modes < E∗(ω)E(ω±νrep) > can be calculated. The intermodebeatnote of the QCL has to be phase-locked to the Local Oscillator of the I/Q Modulator. b SWIFTS correlation spectrumtogether with the spectrum product. Similar to the intermode beat spectroscopy, these results analysed with SWIFTS alsoshows that most of the spectral power is in the frequency comb. c Degree of relative coherence g±(ω, ω ± νrep) obtained bySWIFTS. Results published in63,69.

TABLE I: Comparison between intermode beat spectroscopy and SWIFTS, characterization methods developed for assessingthe relative coherence of QCL combs. (τ : interferometer delay, ν0: reference frequency)

Intermode beat spectroscopy SWIFTS

Measured quantity√S2I (τ, ν0) + S2

Q(τ, ν0)SI(τ, ω0) =< (E(t) + E(t+ τ))2cos(ν0t) >SQ(τ, ν0) =< (E(t) + E(t+ τ))2sin(ν0t) >

Range of ν0 Spectrum analyzer span (centered at νrep) Only a single frequency (usually νrep)Frequency resolution Spectrum analyzer resolution bandwidth Lock-in bandwidth

Sensitivity to incoherent radiation Yes NoPhase retrieval possible No Yes (cummulative sum)

resolution molecular spectroscopy and optical metrologyin the mid-infrared or THz, it is important to assess thefrequency noise characteristics of these type of sources.

With this in mind, the investigation of the frequencynoise power spectral density (FNPSD) of mid-infraredQCL combs was recently performed71. A high-finesseoptical cavity is used to resolve the laser spectrum andto detect the frequency fluctuations of the laser, actingas frequency-to-amplitude (FA) converter, as shown inFig. 17a. The distance between the two mirrors is cho-sen in order to set the free spectral range (FSR) of thecavity close to comb repetition frequency frep. To uti-lize the cavity as a FA converter, a piezoelectric actuator

controlling the cavity length is used and the tempera-ture of the laser is precisely controlled in order to setthe FSR to match exactly the round trip frequency ofthe comb, while simultaneously let the comb offset fre-quency fceo be equal to that of the optical cavity. In thisway, the comb modes and the optical cavity resonancesare perfectly matched. Successfully achieving this align-ment requires an independent control of the fceo and ofthe frep of the QCL comb. As a consequence, in theseconditions and only in these conditions of temperatureand driving current of the laser, all the comb modes aretransmitted by the cavity. The cavity can thus be used asa multimode frequency-to-amplitude converter to collect

13

FIG. 16: Measurement of the deviation from equidistantmode spacing. a Schematic representation of the setup usedfor measuring the deviation from equidistant mode spacingε based on a multi-heterodyne measurement. SA: spectrumanalyzer, BPs: band-pass filters, BS: beam splitter, PI: pro-portional integral, MCT: Mercury cadmium telluride. b Dis-tribution of the deviation from equidistance mode spacingε. Total measurement time = 25.23 s. Gate time = 10 ms.The distribution shows a gaussian distribution with an av-erage value of (−5.6 ± 32) mHz and a standard deviation of558 mHz. Results published in70.

the frequency fluctuations of all the modes at the sametime72. The laser emits a power of 25 mW when the combmodes are exactly matched to the cavity resonances. Aspectrum retrieved with the laser in single-mode opera-tion (P = 15 mW) can also be acquired.

The FNPSD measured on the single-mode and combregimes can therefore be measured by employing thismultimode FA. Fig. 17b shows a magnified view of theFNPSDs in both regimes (shown from 100 kHz to 3 MHz).Around 1 MHz, a flattening characteristic of a white fre-quency noise is observed, corresponding to the intrinsicquantum noise level Dδν due to the spontaneous emis-sion, the so-called Schawlow-Townes limit73. One canalso compare these levels of Dδν to those expected forsingle-mode emission with the same characteristics, givenby the Schawlow-Townes limit74

δν =hν

P

αtotc2

4πn2gαmnsp(1 + α2

e) (13)

Taking ν = 42.2 THz as central frequency, αm =2.2 cm−1 as mirror losses, αtot = 7.2 cm−1 as total cav-ity losses, ng = 3.4, nsp = 2 as spontaneous emissionfactor and 〈α2

e〉 = 0.0023 as squared Henry linewidth en-hancement factor averaged over the laser spectrum, we

FIG. 17: Schawlow-Townes linewidth of QCL combs. a Ex-perimental setup used to measure the FNPSD of the QCLcomb. The main optical components include the QCL laser,the optical isolator, the high-finesse optical cavity and thehigh-sensitivity MCT detector. The signal is processed bya high-sampling-rate oscilloscope. b Zoom of the flatteningportion of the FNPSD of the QCL comb around 1 MHz, cor-responding to the Schawlow-Townes limit. The spectra arecompensated for the FA converter cutoff. The spectra are re-lated to the two operating conditions of the laser: single-modewith P = 15 mW (blue) and comb regime (with all the modesin resonance with the cavity) with P = 25 mW (orange). Re-sults published in71.

can compute the Schawlow-Townes limit relative to thesingle-mode emission (P = 15 mW) and to the combemission (P = 25 mW). The two values are δν = 383 Hzand δν = 230 Hz respectively. These values are consis-tent with those obtained from the spectra δν = πDδν ,which are (474 ± 100) Hz for the single-mode emissionand (292± 79) Hz for the comb emission (see Fig. 17b).

The measurement of the FNPSD in comb regime showsthat the quantum fluctuations of the different modes arecorrelated. In fact, we observe that the FNPSD – in par-ticular the portion limited by the quantum noise – is iden-tical when measured with one comb mode and with allcomb modes simultaneously. This quantum limit, whichis given by the Schawlow-Townes expression, would be atleast a factor of 6 larger than the one shown in Fig. 17b,if we were to assume that the quantum fluctuations ofeach comb mode were uncorrelated. This factor is out-side the uncertainty of the measurement. This experi-mental work demonstrate that the four-wave mixing pro-cess – at the origin of the comb operation in QCLs –correlates the frequency fluctuations between the modes

14

until the quantum limit. As a consequence, instrumentsusing the spectral multiplexing of dual-combs or multi-heterodyne spectrometers hold an inherent noise advan-tage compared to similar systems using arrays of single-mode lasers71.

IV. DISPERSION COMPENSATION

As discussed in the previous sections, a broad gain ofthe QCL ensures low enough GVD to have comb oper-ation. Nevertheless, dispersion is still large enough toprevent the comb from working over the full dynami-cal range of the laser, as can be seen for example inFig. 11 and Fig. 12, and is typical in QCL comb forma-tion54,63,64,69–71. It is therefore of a major interest tocompensate for this dispersion to get comb operationover the entire spectral bandwidth of the laser. In or-der to do such a compensation, it is crucial to have aknowledge of the exact amount of dispersion present inthe laser cavity. Different methods exist for measuringthe dispersion of a QCL. A straight-forward approach isto extract the mode-spacing from a high-resolution FTIRspectrum and calculate the GVD from it. This approachhas proven to give an estimate of the dispersion in THzQCLs64. Another approach is to use THz time-domainspectroscopy to measure the phase difference between apulse that travels once and three times through the lasercavity63. Using this phases one can calculate the GVDof the investigated laser. The theoretically predicted andmeasured dispersion in QCLs is positive, which requiresadditional negative dispersion to compensate for it.

A well-known technique in ultrafast physics to compen-sate for positive dispersion over broad frequency rangesand achiving octave-spanning combs are double-chirpedmirrors (DCM)75. The working principle of a DCM isto delay different frequencies with respect to each otheradding different phases for different frequencies. By care-fully engeneering one can introduce a negative dispersionwhich exactly compensates for the intrinsic dispersion ofthe laser. All that is needed is a sequence of two lay-ers with different refractive index. The thickness of oneperiod fulfills the Bragg condition as in a Bragg mir-ror. The layer thickness is then chirped in a way thatlonger wavelengths penetrate deeper into the DCM andare therefore delayed with respect to shorter wavelengths.In addition, the duty cycle is also chirped to impedancematch the DCM and avoid unnecessary oscillation in theintroduced group delay75.

Terahertz devices Recent work introduced DCMs tocompensate the dispersion in THz QCLs63. Since the di-mension of DCMs is given by the Bragg condition suchstructures can be fabricated at THz frequencies by usingstandard microfabrication techniques. Instead of usingtwo materials with different refractive index, the waveg-uide width has been tapered in the implementation re-ported in ref.63. Since the effective index of the waveg-uide is strongly dependent on the width, such approach

is equivalent to using two media with different refrac-tive indeces. This scheme is shown in Fig. 18a. Start-ing and stoping period of the tapering are defining thefrequency range covered by the DCM. The corrugationlength is defining the amount of dispersion introducedby the DCM. The optical spectrum obtained with suchdevice is shown in Fig. 15b63.

A similar approach can be used to compensate for thedispersion of an octave-spanning THz QCLs. A first ver-sion has recently been realized. In order to fully exploitthe technique of DCMs, sections of benzocyclobutene(BCB) and semiconductor active material have been al-ternated to get a higher refractive index contrast thanjust corrugating the waveguide as in ref.63. The top met-allization is kept continuous over the entire structure toget a good mode confinement. A series of such lasers hasbeen realized with this technique (Fig. 18b). As shown inFig. 18c,d, a narrow beatnote is observed on such devicesindicating comb operation over a bandwidth of 500 GHz.Further optimization on the design of the DCM shouldallow to also compensate for a full octave in frequency.

FIG. 18: Double-chirped mirror architectures for THz QCLs:a Approach published by the MIT group in ref.63: The waveg-uide width is varied to achieve a contrast in refractive index.Both period and amplitude are chirped to introduce anoma-lous dispersion over a large frequency range. The longer thecorrugation length the more dispersion is added. The rightfigure shows an SEM picture of a fabricated structure fromref.63 b Our approach, where the mirror is realized by inter-secting sections of BCB and active region (MQW) to achievea high refractive index contrast. c Beatnote and d corre-sponding optical spectrum of a DCM (ETHZ design) in CWoperation at 15 Kelvin indicating comb operation over a spec-tral bandwidth of 500 GHz.

Mid-infrared: Gires-Tournois coatings Another con-cept that can be implemented to compensate for dis-persion is the use of a Gires-Tournois Interferometer76

(GTI) directly integrated into the QCL comb. In con-trast to DCMs, it can only compensate a limited fre-quency range but gives more flexibility on the amount ofdispersion that can be introduced. This method was re-

15

cently employed for the dispersion compensation of mid-infrared QCL combs, by directly evaporating the GTIon the back-facet of a QCL comb65. By controlling thedispersion, the range where the comb operates signifi-cantly increases, effectively suppressing the high-phasenoise regime usually observed in QCL combs54,63,69–71.In particular, the comb regime was observed over the fulldynamical range of operation of the device up to powerslarger than 100 mW.

V. APPLICATIONS: QCL-BASED DUAL-COMBSPECTROMETERS

While frequency combs can be used for spectroscopyby combining them with dispersive elements4, the mostattractive use of these devices is clearly the multi-heterodyne or dual-comb configuration10,11, because itleverages on the unique coherence properties of the comb.As schematically described in Fig. 2 and as shown in inFig. 19c for an implementation using QCL combs, dual-comb spectroscopy is based on the measurement of abeating created by two frequency combs with slightly dif-ferent repetition frequencies (frep,1 and frep,2 = frep,1 +∆f , respectively, where ∆f is the difference in the combsrepetition frequencies).

Results using this technique combined with QCLcombs, published in ref.70, are summarized in Fig. 19.The optical spectrum of two QCL combs tuned to allowmultiheterodyne spectroscopy is shown in Fig. 19 a, whilethe RF spectrum associated with the beating of two QCLcombs is shown in Fig. 19b. These key results shows theprinciple of dual-comb spectroscopy. A detector is usedto measure the multi-heterodyne beat, corresponding toFig. 19b. Each line of the first comb will beat with alllines of the second comb creating several different beat-notes in the RF domain. The difference in repetitionsfrequencies ∆f must be chosen such as to obtain a one-to-one mapping between the lines of the two combs andtheir RF counterpart.

In one form of dual-comb spectroscopy77, one comb isused as a local oscillator while the other is used to inter-rogate a sample, as shown in Fig. 19c. Due to the one-to-one mapping, each multi-heterodyne beat contains in-formation regarding the sample absorption at the opticalfrequency of the comb line interrogating the sample. Asthe technique relies on the discrete nature of a frequencycomb, the sample absorption is measured at frequen-cies spaced by the comb repetition frequency (7.5 GHz= 0.25 cm−1 for the example of Fig. 19), thus definingthe resolution of the dual-comb spectrometer. For theinvestigation of large organic molecules in gas phases orfor the study of liquids, this resolution is usually suffi-cient. However, for gas spectroscopy of small moleculesat low pressures, the linewidth of molecule absorptionlines are usually narrower (hundreds of MHz) than QCLcomb repetition frequencies, usually ranging from 5 GHzup to 50 GHz for conventional devices. A special fea-

FIG. 19: Dual-comb spectroscopy based on QCL combs:a Typical optical spectra of two mid-infrared QCL combs,where the offset frequencies fceo of both combs are identical.b Multi-heterodyne beat of two QCL combs with slightly dif-ferent comb repetition frequencies measured on a fast detec-tor. The multi-heterodyne beat signal contains informationon the sample absorption. c Schematic view of the dual-comb spectroscopy setup based on QCL frequency combs.One comb is used as a local oscillator (LO) while the otherinterrogates the gas cell. BS: 50-50 Antireflection coatedbeam splitter, NDF: neutral density filter. d Water vapourtransmission spectra in air (PH2O = 1.63 kPa, total pressurePtot = 101.3 kPa, T = 20C) measured with our dual-combspectrometer (800 MHz of spectral resolution after averaging)and HITRAN simulation. Inset: expansion of the measuredtransmission over ∼1 cm−1 with the full resolution (80 MHz).Results published in70.

ture of QCL combs is that the comb teeths can be sweptover an entire frep, by tuning either the laser tempera-ture or current, therefore increasing the resolution of theQCL-based dual-comb spectrometer.

A dual-comb spectrometer based on QCL combs is de-picted in Fig. 19c. The difference in repetitions frequen-cies ∆f can be set between 5 to 40 MHz by changingthe temperature and current of both combs. A dual de-tection technique is implemented as it helps to removetechnical noise on the detected amplitude78. As a proofof principle applied to gas sensing, transmission measure-ment of water vapour in air at atmospheric pressure in a6 cm-long gas cell is measured. We applied a frequencysweep to the combs in order to achieve high-resolution(80 MHz step over a bandwidth equal to one comb rep-etition frequency). In order to avoid parasitic fringescoming from residual reflectivities in the beam path, themulti-heterodyne beat signal is measured with the gascell filled with water vapour and subsequently with ni-trogen, at each step of the frequency sweep. A referencemeasurement was thus taken at each step of the sweepand used to deduce the absolute value of the transmis-sion.

Fig. 19d shows the transmission of water vapour in air

16

(partial pressure of water vapour in air PH2O = 1.63 kPa,total pressure Ptot = 101.3 kPa, T = 20C) measuredwith the dual-comb setup as well as a transmission simu-lation using HITRAN database79. The wavenumber scalecalibration was done by using the HITRAN simulationand by applying a rigid shift to the measured transmis-sion spectrum in order to fit the HITRAN simulationspectrum. Over the entire duration of the measurement(few hours), the slow drifts reduce the quality of the in-terleaved transmission spectrum and a moving averagefilter was used for smoothing the interleaved transmissionspectrum, reducing the effective resolution to 800 MHz.

A clear agreement between the two measurementsis observed over the entire measurement bandwidth(16 cm−1) and the absolute value of the transmissioncould also be retrieved. Due to the important attenu-ation of the comb lines situated in the vicinity of thewater absorption lines, their attenuated amplitude couldlie below the noise floor of the detection setup. In sucha case, the algorithm calculating the absorption will notbe able to retrieve the entire shape of the absorptionline. This can be observed on the shape of the waterabsorption line situated at 1419.5 cm−1. Finally, as themulti-heterodyne spectrum presents beatnotes with dif-ference in amplitudes of more than 40 dB, the very lowintensity beatnotes will not be detected by the algorithmcalculating the absorption. This results in some holes onthe spectrum and can also be observed on Fig. 19c, forinstance close to the water absorption line situated at1416.3 cm−1.

Frequency (MHz)300 400 500 600 700 800 900 1000 1100

Ampl

itude

(a.u

.)

106

107

108

109

- 0.9Ampl

itude

(a.u

.)

(MHz+554 MHz)+ 0.90 - 0.9 + 0.90 - 0.9

(MHz+971 MHz)(MHz+971 MHz)+ 0.90

Most intense peak

fceo correction

fceo and frep correction

FIG. 20: Digital adaptive sampling technique used to correctfceo and frep fluctuations. The improved algorithm correctsfor both fceo and frep.

In general, the signal over noise characteristics ofthe absorption measurements in free-running dual-combspectroscopy reflects a trade-off between the spectralbandwidth, number of comb lines and amplitude accu-racy. Using conventional Fabry-Perot QCL devices, dual-comb spectroscopy was demonstrated successfully on arelatively limited spectral bandwidth80. In fact, the ob-servation of the Schawlow-Townes limit on the linewidthof QCL combs proves that the comb formation processdoes not add additional mode partition noise; what mustbe done is to remove the fluctuations in the fceo and frep

in both combs. In conventional dual-comb systems, suchnoise has been efficiently removed by ”adaptative sam-pling” techniques81.

Such adaptive sampling techniques can be fitted toQCL combs. The results presented in Fig. 19d wereachieved by correcting fluctuations in fceo but neglectedthe fluctuations of frep. The spectra were aligned spec-trally along the main peak; as a of fluctuations of the frep,the wings of the spectra suffered from a conversion of thatfrequency noise into an additional amplitude noise. Thiseffect is shown in Fig. 20: while the fluctuations of themulti-heterodyne beatnote at around the most intensepeak (∼ 554 MHz) are corrected over the whole acquisi-tion time, the peak at (∼ 971 MHz) is fluctuating by morethan its linewidth. When frep is simultaneously retrievedand the signal resampled, the frequency fluctuations aresignificantly reduced.

The effect of this correction of both fceo and frep isshown in Fig. 21 where two subsequent acquisition areratioed, as an indication of the instrument baseline. Themeasured SNR of each measurement point varies accord-ing to the relative intensity of the peak. In an acquisi-tion time of 25 ms, SNR of ¡1×10−3 of high power peaksare reached whereas low power peaks exhibit a SNR of¡4× 10−2. Compared to the situation where only fceo iscorrected, the useful bandwidth of the dual-comb spec-trometer is increased from 16 to 45 cm−1.

Wavenumbers (cm -1)5 10 15 20 25 30 35 40 45

Tran

smis

sion

0.96

0.97

0.98

0.99

1.00

1.01

1.02

1.03

1.04

FIG. 21: Baseline measurement of dual-comb spectrometerwith both fceo and frep corrections. Data acquisition time is25 ms.

Similar, preliminary results were recently presented,demonstrating the use of correction techniques to THzcombs? . As such, although self-referencing is a desir-able property of such combs ultimately, it is probably nota requirement for the development of high performancesystems.

VI. CONCLUSION AND OUTLOOK

As compared to mode-locked monolithic semiconduc-tor lasers, QCL combs benefits from features specific tothe physics of its intersubband gain medium: the capa-bility to achieve very wide gain bandwidths, the low ma-terial GVD as well as the FM-like characteristics of the

17

emission that allows large average powers without highpeak powers incompatible with tightly confined waveg-uides. The possibility to build broadband spectrometerswith chip-sized, electrically pumped optical frequencycomb sources has a great potential for applications insensing and spectroscopy. QCL combs have, in addi-tion, some additional potential features that have notyet been exploited: because of the ultrafast transport inthe active region, QCLs are also RF detectors that arein principle capable of detecting a heterodyne beatnote.The recent progress in broadband gain active regions inthe mid-infrared and THz show that an octave-spanningQCL comb is feasible. In addition, the active region canalso be engineered to provide a χ(2) susceptibility, and

therefore internally generate its own second harmonic.This capability of the QCL comb to integrate gain, non-linearity as well as detection is unique and offers greatpotential for device and system integration.

VII. ACKNOWLEDGEMENT

The authors want to acknowledge the support fromthe Swiss National Science Fundation, the ETH pioneergrant, the FP7 project TERACOMB as well as from theDARPA program SCOUT.

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