High-resolution spectroscopy of arbitrary light sources using frequency combs
HHigh-resolution spectroscopy of arbitrary light sources using frequency combs
David Burghoff
Department of Electrical Engineering,University of Notre Dame, Notre Dame, IN 46656 (Dated: April 23, 2020)Multiheterodyne techniques using frequency combs—light sources whose lines are perfectly evenly-spaced—have revolutionized optical science. By beating an unknown signal with the many lines ofa comb, its spectrum is recovered. However, these techniques have been restricted to measuringcoherent sources, such as lasers. In this work, we demonstrate a new multiheterodyne techniquethat allows for nearly any complex broadband spectrum to be retrieved using a comb. Two versionsare introduced: a delayed comb technique that uses a tunable delay element, and a dual combtechnique that uses a second comb. In each case, the spectrum of the source is recovered by Fouriertransforming the correlation between two spectrograms. This approach is statistical in nature and isgeneral to nearly any source (coherent or incoherent), allowing for the entire spectrum to be rapidlymeasured with high resolution.
I. INTRODUCTION
A fundamental problem in optical science is detectingthe spectrum of a remote source. Heterodyne detection isa powerful technique for performing high-resolution spec-troscopy. By beating an unknown optical signal from aremote source with a known local oscillator (LO), onecan measure the mixing between the two to achieve ahigh-precision measurement of the unknown signals spec-trum. As this technique is particularly relevant for mea-suring the spectra of distant objects, it has found sig-nificant use in astronomy. However, a limitation of thistechnique is that it can only measure spectra over a lim-ited bandwidth, effectively covering only frequencies thatare within the mixers intermediate frequency (IF) band-width. This restriction can be overcome by requiringwidely tunable local oscillators, but tuning can be diffi-cult when the LO is an optical source, as tunable LOstypically require moving parts.A potentially attractive alternative is the use of fre-quency combs, broadband light sources whose lines areperfectly evenly spaced. Frequency combs have enableda wide variety of multiheterodyne techniques , which in ef-fect use multiple LO lines to detect multiple signals [1].For example, dual comb spectroscopy [2–8] can be used tomeasure the spectrum of another comb, comb-referencedapproaches can measure the spectrum of a laser [9, 10],and vernier spectroscopies [11–13] can be used to mea-sure the spectra of multi-line lasers. However, all of thesetechniques require that the resulting spectra do not over-lap at intermediate frequencies (IFs), as overlapping IFscreate an unavoidable ambiguity in the spectrum. Thisprecludes their use in applications where the optical sig-nal is broadband and may even be broader than the combspacing—remote sensing, astronomy, biological systems,etc.We introduce a new high-resolution multiheterodynetechnique that is able to unravel the spectrum of arbitrary light sources, even incoherent sources whose linewidthsare much greater than the comb spacing. Inspired by the interferometer-based techniques that disambiguate indi-vidual comb lines [14–18], we show that a signal mixingwith a comb can be fully disambiguated using a vari-able delay element, even when multiple signals appearat the same IF. We also show that a dual comb versionof this approach can accomplish the same task, rapidlymeasuring the spectrum of any source without movingparts. Each version of this measurement has analoguesto Fourier spectroscopy and preserves many of its fea-tures, such as the throughput and multiplex advantages[19, 20]. Though the approach relies on the comb struc-ture, it does not require that the combs have a particularphase profile—the combs can be pulsed [11, 13, 21, 22]or not [17, 21, 23–28].
II. OVERVIEW
A simplified version of the requisite experimental se-tups are shown in Figure 1. In each measurement, thesource to be measured is split and is mixed with twocombs. In the dual comb version, these two combs comefrom separate sources. They may or may not be mutu-ally coherent. In the delayed comb version, the secondcomb is generated from the first by using a variable de-lay element. Essentially, it is a Doppler-shifted comb [14].In each case, the detector signals are digitized and pro-cessed into complex spectrograms. (This is done by di-viding the data into batches and computing a short-timeFourier transform.) The product of the two spectrogramsis computed, and the result is then Fourier transformedagain to achieve the final result. Even for a fully incoher-ent source this correlation function is proportional to thepower of the signal (offset from n th comb line), which al-lows it to reconstruct essentially any source. The detailsof the two versions differ only slightly.In the dual comb version, the complex spectrograms F i ( ω, T ) are functions of the IF frequency ω and the timeof each spectrogram T . We denote the respective posi-tion of the n th comb lines as ω ( c ) n and ω ( c ) n ≡ ω ( c ) n +∆ n , a r X i v : . [ phy s i c s . op ti c s ] A p r Comb 1 Comb 2signal slow time T I F ω F ( ω ,T) slow time T I F ω F ( ω ,T) ω n (c ) ω n (c ) = ω n (c ) + Δ n Combsignal I F ω F ( ω , ) I F ω F ( ω , ) ω n delay speed v (c) delay delay Dual comb version Delayed comb version
Figure 1. Overview of the two approaches. In the dual combversion two combs are independently beat with the signal; inthe delayed comb version one comb is split and delayed. Bothdetectors’ spectrograms are computed and are correlated toreproduce the original signal’s spectrum to high resolution(the measurement time divided by the number of comb lines). where ∆ n is the separation between corresponding lines.We denote the respective complex amplitudes as E ( c ) n and E ( c ) n . We then correlate the two spectrograms andcompute C n ( ω ), the Fourier transform of the spectro-gram correlation along T , C n ( ω ) ≡ F T [ F ( ω − ∆ n , T ) F ∗ ( ω, T )]( − ∆ n ) . As we show in Appendix A, this function is statistically-related to the spectrum of the source P s by (cid:104) C n ( ω ) (cid:105) = E ( c ) ∗ n E ( c ) n P s ( ω ( c ) n + ω ) . (1)In other words, the spectrum near a comb line can bedetermined simply by dividing out the amplitude of thedual-comb beat signal E ( c ) ∗ n E ( c ) n . This result holds forboth positive and negative IF frequencies as well as over-lapping IF frequencies, allowing for complete disambigua-tion of the signal.The delayed comb version is similar. Here the complexspectrograms F i ( ω, τ ) are functions of delay τ , and aretypically related to laboratory time by τ = vc T (providedthe delay element moves at a constant velocity v ). Inthis case there is only one set of comb frequencies andamplitudes, and the correct definition for C n is similar: C n ( ω ) ≡ F τ [ F ( ω + vc ω ( c ) n , τ ) F ∗ ( ω, τ )]( ω ( c ) n ) (cid:104) C n ( ω ) (cid:105) = E ( c ) ∗ n E ( c ) n P s ( ω ( c ) n + ω ) . (2)Once again, the spectrum near a comb line can be de-termined by dividing out the corresponding comb tooth power, in this case | E ( c ) n | . Detailed derivations are givenin Appendix A.Both of these approaches are extremely general, re-constructing the source in practically all cases. The solesituation in which they will not correctly reproduce thespectrum of the signal is when there exist frequencies forwhich (cid:104) E s ( ω ) E ∗ s ( ω + nω r ) (cid:105) (cid:54) = 0 over the duration of themeasurement (where ω r is the repetition rate of a comband n is an integer). Over sufficiently long timescales,this will only fail when the source under consideration isdeliberately chosen to match the combs’ repetition rates,for example by attempting to measure another comb withthe same spacing. The approach is also general for alltypes of combs, irrespective of the phase of the comblines. III. RESULTS
As a relevant example, we consider a complex terahertzspectrum consisting of several lines, similar to the typeof signal that is highly relevant for astronomy (for exam-ple, in measuring the spectral line energy distribution ofcarbon monoxide [29]). We consider signals in the rangeof 4 to 5 THz and consider the dual comb version of themeasurement. Our combs are assumed to span 4-5 THzwith repetition rates of 10 GHz (typical parameters forquantum cascade laser combs [16, 30]). Our signals arefairly broadband—with 100 MHz full-width half maxi-mums (FWHMs)—and are generated numerically usinga phase random walk process. A list of the line strengthsand locations is shown in Table I.These lines are chosen to illustrate the power of thetechnique. Lines A, C, and D appear at positive IFs(relative to the nearest comb line), while line B appears ata negative IF. With a single LO line and one detector, it isimpossible to distinguish positive IFs from negative IFs.Furthermore, lines C and D appear at the exact same IF,which means that distinguishing them is impossible withall prior vernier-like techniques. In this case, the linesare relatively broadband but are much narrower than thecomb spacing. The corresponding magnitude of the twospectrograms is shown in Figure 2a. Individually, the two
Line Frequency(GHz) Power(pW) Offsetfrom comb1 (GHz) Offsetfrom comb2 (GHz)A 4211 0.1 1 0.669B 4558 0.225 -2 -2.366C 4783 0.4 3 2.612D 4803 0.625 3 2.610Table I. Lines of the spectrum considered. Comb 1 spans4-5 THz with a repetition rate of 10 GHz. Comb 2 has arepetition rate of 10 GHz+1 MHz and has an additional offsetof 0.3 GHz. Comb lines have a power of 1 mW per tooth.
Time (ms) F r e q u e n c y ( G H z ) -40-200 P o w e r ( d B ) Time (ms) -5 F r e q u e n c y ( G H z ) -40-200 P o w e r ( d B ) |F ( ω ,T)| |F ( ω ,T)| Raw spectrogramRaw power spectral densities
C, DBA
Reconstructed spectrum
AB CD actualreconstructed residual abc <|F ( ω )| ><|F ( ω )| > -5 -4 -3 -2 -1 0 1 2 3 4 5 Intermediate frequency (GHz) P S D ( n W / M H z ) -0.200.2-0.200.2-0.200.2-0.200.2 -5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.6-5 -4 -3 -2 -1 0 1 2 3 4 500.51-5 -4 -3 -2 -1 0 1 2 3 4 5012-5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency (GHz)Offset from nearest comb 1 line
Offset from4220 GHz Offset from4560 GHz Offset from4780 GHz Offset from4800 GHz P o w e r s p e c t r a l d e n s i t y ( n W / M H z ) R e s i d u a l ( n W / M H z ) Figure 2. a. Magnitudes of the recorded spectrograms asa function of slow time and IF frequency (10 MHz RBW,0.45 ms measurement time). b. Raw signal power spectraldensities, with contributions from beating with various lineslabeled. c. Reconstructed signals calculated from equation(1), along with the actual spectrum. spectrograms appear as a noisy version of the averagepower spectral density of the signal on the two detectors.This ’noise’ actually arises from the incoherent nature ofour signals, and it is the hidden correlations of these twosignals that give rise to our computed result. Similarly,the average power spectral densities of the two signals areshown in Figure 2b. They contain peaks from all linesbeating with the combs—the result is a complex zoo ofoverlapping spectra. Figure 2c shows the results of our correlation calcula-tion. For each comb line, we compute the real part of C n ( ω ) and plot the result for comb lines that are nearsignal lines. We also show a theoretical prediction of (cid:12)(cid:12)(cid:12) E ( c ) ∗ n E ( c ) n (cid:12)(cid:12)(cid:12) P s ( ω ( c ) n + ω ). The agreement between thetwo is excellent. For example, looking at the spectrumnear line A (the weakest line), very little evidence of otherlines is present despite the fact that much larger signalsare beating at ± ± F and F must be conjugate-symmetric, C n is not. Finally, note that lines C and Dare also correctly distinguished despite the fact that theyoverlap entirely with respect to both combs 1 and 2.These results are valid for any source, even extremelyincoherent sources. To illustrate this, we increase theFWHM of the lines shown above to 10 GHz—the combrepetition rate—and plot the reconstructions. Withlinewidths this broad, the entire IF span is filled witha relatively flat spectrum (see Figure 3a). Since the truespectrum does not have narrow features, we plot the fullreconstructed spectrum in Figure 3b. Once again, theresults are in good agreement with the theoretical pre-diction. Note that because lines C and D are only 2 f r apart, their lineshapes overlap both in real frequency andin intermediate frequency. Although in this case there isvery little information to be gained from going to reso-lution bandwidths this narrow, in the general case wherethe spectrum’s features are totally unknown the abilityto perform high-resolution measurements is critical. IV. DISCUSSION
As this technique extends multiheterodyne spec-troscopy to the case of incoherent spectra, it maintainsmany of the appealing features of other multiheterodynetechniques. The dual comb version is similar to otherdual comb techniques in that the ultimate time resolu-tion of this measurement is determined by the abilityto resolve different dual comb beat signals [11, 12]. Assuch, its time resolution is 1 / ∆ f r and can easily be withinthe range of microseconds (contingent on the signal-to-noise ratio). While the delayed comb version is limitedby the mechanical speed of the delay element, chip-scalecombs such as quantum cascade lasers and microres-onator combs typically have repetition rates in the tenof GHz range, meaning that resolving their features re-quires only a travel distance of a few millimeters. This isconceivably within the range of 100 ms.Next, we consider resolution. For perfect comb sourcesthe resolution bandwidth (RBW) is determined by thebatch length according to RBW= N t ∆ t , where ∆ t is thesample rate and N t is the number of samples per batch.However, resolving the dual comb beat term requires thatthere be at least as many batches ( N T ) as the number ab -2024000 4200 4400 4600 4800 5000 Frequency (GHz)
Raw power spectral densityReconstructed broadband spectrum <|F ( ω )| ><|F ( ω )| > -5 -4 -3 -2 -1 0 1 2 3 4 5 Intermediate frequency (GHz) R e s i d u a l ( n W / M H z ) -3 P o w e r s p e c t r a l d e n s i t y ( n W / M H z ) P S D ( n W / M H z ) actualreconstructedresidual Figure 3. Reconstruction of a spectrum whose componentsare broader than the comb spacing. a. Raw power spectraldensities of the recorded signals. They are essentially flat,having no discernible peaks. b. Spectral reconstruction andtrue spectrum (10 MHz RBW, 9 ms measurement time). of comb lines ( N c ). If T is the total measurement time T ≡ N T N t ∆ t , then this requires that RBW > N c T . Inother words, the resolution is limited by the measure-ment time divided by the number of comb lines. This isworse than what is achievable with isolated lines (whichallows for resolutions of 1 /T [11]) because knowing thatthe line is isolated requires prior knowledge of the sig-nal. Effectively, we traded some resolution for the abilityto distinguish lines that overlap in the IF. Practicallyspeaking this is only relevant for phase-stable combs; forfree-running combs it is typically the linewidth of thecombs themselves that set the technique’s resolution.In terms of sensitivity, this approach has many similar-ities to traditional Fourier spectroscopy. Because the en-tire signal is measured at once and is not demultiplexed,the multiplex advantage [19] is maintained. Addition-ally, because the system does not require a single-modesource, the throughput advantage [20] is maintained. Adetailed discussion is given in Appendix B; for additivenoise, the total noise is the sum of image noises fromevery signal sharing an IF frequency. When comparedwith a tunable LO of the same total power, in bestcase scenario of non-overlapping signals the sensitivitiesare identical. In the worst case scenario of broadbandspectra the RMS sensitivity is √ N c worse, as the LOpower of the individual lines is effectively divided by N c .While this is problematic for certain astronomical ap-plications where quantum-limited sensitivity is required,for dynamically-varying sources the dual comb version ofthis approach will be considerably faster than a widely- tunable LO (which typically requires moving parts) or aFourier spectrometer. Detector nonlinearity will furtherdegrade the sensitivity, see Appendix D.Like all comb-based spectroscopies, this approach suf-fers in the presence of comb phase noise. While the de-layed comb version is fairly immune to such effects, thedual comb version is highly susceptible. Still, like dual-comb spectroscopy [31–36], it can be corrected after thefact. This procedure is detailed in Appendix C, and es-sentially requires that a standard dual comb measure-ment (measuring the beating of the two combs) is per-formed. This additional measurement can also be usedto calibrate the amplitude, as a dual comb measurementwill produce the beat signal E ( c )) n E ( c )) ∗ n that is neededto normalize the result.Lastly, we place this result in the context of earlierwork, in particular with respect to vernier spectroscopy[11–13]. While the delayed comb version of this ap-proach has not been demonstrated in any capacity (to ourknowledge), the dual comb version is experimentally verysimilar to vernier spectroscopy. Previous work did notattempt to perform spectroscopy of incoherent sources,focusing on swept-source diode lasers [11–13] and fibermode-locked lasers [13]. The key distinction is that in allof these cases, individual lines can be isolated providedwhen they do not occur at the same IF frequency, andcorrelation maximization procedures can be used to findthe comb order number of each line. However, this ap-proach does not generalize to arbitrary incoherent spec-tra, as continuous overlapping spectra from different or-ders are not compatible with this approach. Still, equa-tion (1) is immediately applicable to these results, pro-viding a new way to analyze them. V. CONCLUSION
We have shown theoretically and numerically that fre-quency combs can be used to unravel the spectrum of anarbitrary incoherent light source. This can be done eitheruse two separate combs or one comb with a copy that hasbeen delayed. Even when the beating of the two combsappears random and chaotic, a periodically-varying cor-relation persists in the complex spectrograms, and thiscan be exploited to infer the spectrum. This result iscompatible with all existing comb technologies and willallow for passive high-speed spectroscopy of essentiallyany dynamically-varying electromagnetic source. For ex-ample, one can imagine applications in reaction kinetics[37], in biology [38, 39], and even in millimeter-wave sys-tems [40].
Appendix A: Detailed derivation
To show that equations (1) and (2) hold, we will derivethe result for the dual comb version first and extend it tothe delayed comb version. All electric fields are expressedas a superposition of exponentials as E c i ( t ) = (cid:88) n E ( c i ) n e iω ( ci ) n t and E s ( t ) = (cid:88) m E ( s ) n e iω ( s ) m t where E c i ( t ) is the field of the ith comb and E s ( t ) isthe field of the signal to be measured. For conveniencethe signal is represented as a summation rather than anintegral. In a heterodyne measurement, the raw signalthat is recorded is S i ( t ) = 12 | E c i ( t ) + E s ( t ) | ∼ E ∗ c i E s = (cid:88) n,m E ( c i ) ∗ n E ( s ) m e i ( ω ( s ) m − ω ( ci ) n ) t where we neglect the intracomb beat terms (which oc-cur only at vanishingly-narrow multiples of the repetitionrate) and the intrasignal beat terms (which are assumedto be small).We assume that our detector has a bandwidth largerthan f r / t sufficientlysmall to avoid aliasing. In the spirit of spectrograms,time is divided into two separate time axes—the fasttime t i and the slow time T j —such that the total timeis given by t = t i + T j . The data is similarly dividedinto batches that are N t ∆ t long (which determines theresolution bandwidth). In other words, T j can be takenas T j = ( j − N t ∆ t , where j is an integer. Dual comb version.
First, we calculate the sig-nals in terms of the FFT IF frequencies ω k ≡ πN t ∆ t k .For the FFT, we use the convention that F [ f ]( ω k ) ≡ N t (cid:80) i e − iω k t i f i . For comb 1, we find that the spectro-gram F ( ω k , T j ) is given by F ( ω k , T j ) = 1 N t (cid:88) i e − iω k t i S ( t i + T j )= 1 N t (cid:88) i,n,m E ( c ) ∗ n E ( s ) m e i ( ω ( s ) m − ω ( c n − ω k ) t i × e i ( ω ( s ) m − ω ( c n ) T j . Because our resolution bandwidth is determined by ourbatch length (RBW= N t ∆ t ), in order to proceed wemake an approximation in which our signal frequenciesare all an integer number of RBWs away from comb 1,i.e. ω ( s ) m − ω ( c ) n = πN t ∆ t l for some integer l . As a result, e i ( ω ( s ) m − ω ( c n ) T j = 1, and F ( ω k , T j ) = (cid:88) n E ( c ) ∗ n N t (cid:88) i e − i ( ω ( c n + ω k ) t i × (cid:88) m E ( s ) m e iω ( s ) m t i = (cid:88) n E ( c ) ∗ n E s ( ω ( c ) n + ω k ) , where in the last line we used the definition of the FFTtwice. Due to our finite resolution bandwidth approxi-mation, the spectrogram would appear to be constant in T . However, once this approximation has been made it cannot be modified when computing the same quantityfor comb 2. Performing a 1 → e i ( ω ( s ) m − ω ( c n ) T j = e i ( ω ( s ) m − ω ( c n − ∆ n ) T j = e − i ∆ n T j F ( ω k , T j ) = (cid:88) n E ( c ) ∗ n E s ( ω ( c ) n + ω k ) e − i ∆ n T j . Thus, while F is stationary in time, F is not, and infact beats periodically at the dual comb frequencies ∆ n .Provided the data has been recorded long enough to re-solve individual beat frequencies (i.e., that the data isat least recorded for 1 / ∆ f r = 1 / | f r − f r | ), these beat-ings can be resolved. For simplicity, we assume that thenumber of batches N T has been chosen to ensure that N T N t ∆ t = 1 / ∆ f r , which will resolve exactly one dualcomb beat tooth.Finally, we compute the Fourier transform of the cor-relation function and its expectation value: C n ( ω k ) ≡ F T [ F ( ω k − ∆ n , T j ) F ∗ ( ω k , T j )]( − ∆ n ) (cid:104) C n ( ω k ) (cid:105) = 1 N T (cid:88) j,l,m E ( c ) ∗ l E ( c ) m e i (∆ n − ∆ l ) T j × (cid:68) E s ( ω ( c ) l + ∆ l − ∆ n + ω k ) E ∗ s ( ω ( c ) m + ω k ) (cid:69) Because the number of batches was chosen to be an inte-ger number of 1 / ∆ f r , the summation N T (cid:80) j e i (∆ n − ∆ l ) T j is a summation over roots of unity and vanishes unless n = l , leaving (cid:104) C n ( ω k ) (cid:105) = (cid:88) m E ( c ) ∗ n E ( c ) m × (cid:68) E s ( ω ( c ) n + ω k ) E ∗ s ( ω ( c ) m + ω k ) (cid:69) (A1)This result is general for any source. For the vast ma-jority of sources we can make an additional assumption,which is that the long-term correlation between compo-nents of frequencies spaced by the repetition rate of thecomb vanishes. This assumption is essentially valid forany source that is not a comb of the same spacing as ei-ther of the LO combs. With this additional assumptionwe can eliminate the cross terms, leaving (cid:104) C n ( ω k ) (cid:105) = E ( c ) ∗ n E ( c ) n (cid:28)(cid:12)(cid:12)(cid:12) E s ( ω ( c ) n + ω k ) (cid:12)(cid:12)(cid:12) (cid:29) . (A2)By calculating every C n , the signal’s power relative to acomb line can always be extracted. Delayed comb version.
For the delayed comb, the anal-ysis is similar. The analysis for comb 1 is fully identical(letting E ( c ) n → E ( c ) n ), and the analysis for comb 2 isfound by setting E ( c ) n → E ( c ) n e − iω ( c ) n τ and ∆ n = 0. How-ever, a subtlety arises when delay is a linear function oflab time (i.e., a linear scan is performed rather than astep scan). Because delay changes during the batch, thishas the same effect as Doppler shifting the IF frequencies -5 -4 -3 -2 -1 0 1 2 3 4 5 Offset frequency (GHz) -7 -6 V a r i a n c e ( n W / M H z ) Variance of C n ( ω ) analytical Monte Carlo Figure 4. Variance of the reconstruction in Fig. 2 for whitenoise of RMS power 10 nW, evaluated analytically using (B1)and numerically using Monte Carlo. in a manner similar to a nonzero ∆ n . One should there-fore proceed as before, but using the explicit mapping τ → vc ( t i + T j ). This results in F ( ω k , T j ) = (cid:88) n E ( c ) ∗ n E s (cid:18) ω ( c ) n − ω ( c ) n vc + ω k (cid:19) e iω ( c ) n vc T j By comparing this to the dual comb version of the sameresult, we find that − ω ( c ) n vc has replaced ∆ n . If we nowdefine τ j ≡ vc T j and make the appropriate substitutionsinto our definition of C n , we find that we must insteadcalculate C n ( ω k ) ≡ F τ [ F ( ω k + ω ( c ) n vc , τ j ) F ∗ ( ω k , τ j )]( ω ( c ) n )which results in (cid:104) C n ( ω k ) (cid:105) = (cid:12)(cid:12)(cid:12) E ( c ) n (cid:12)(cid:12)(cid:12) (cid:28)(cid:12)(cid:12)(cid:12) E s ( ω ( c ) n + ω k ) (cid:12)(cid:12)(cid:12) (cid:29) . Appendix B: Sensitivity
For astronomical applications, it is important to ana-lyze the sensitivity of this approach. We do so for thecase of additive white Gaussian noise (AWGN) and com-pared with the sensitivity of a tunable local oscillator.We assume that each detector measurement S i ( t ) is per-turbed by a white noise source with variance σ . Thisnoise source is also taken to be uncorrelated between eachdetector. By propagating this noise through the recon-struction, one can show that the variance of the extractedsignal in the dual comb version is given byVar [Re C n ( ω )] = 12 N T (cid:18) σ N t (cid:19) + σ N T N t (cid:88) m (cid:12)(cid:12)(cid:12) E ( c ) m (cid:12)(cid:12)(cid:12) P s ( ω ( c ) m + ω )+ (cid:12)(cid:12)(cid:12) E ( c ) m (cid:12)(cid:12)(cid:12) P s ( ω ( c ) m − ∆ n + ω )(B1)As an example, in Fig. 4 we plot this expression forthe data plotted in Fig. 2. Because the summationover m is over both positive and negative frequencies, this means that the noise at a single frequency comesfrom the double-sided power spectral density of all signalcomponents that share an IF with the frequency underconsideration. The noise is frequency-dependent and re-sembles the sum of the two raw power spectral densities.This is practically speaking a dynamic range limitation:if one attempts to measure a weak signal that shares anIF with a much stronger signal, image noise generated bythe stronger signal will swamp the weaker one. Whetherthis is tolerable depends on the source and on the appli-cation. For spectra consisting of relatively narrow linesit can be avoided by a proper choice of f r , for example.The corresponding expression for using a single LOwith electric field E e iω t + c.c. to measure double side-band intensity isVar [ I ( ω )] = 1 N T (cid:18) σ N t (cid:19) + 2 σ N T N t | E | ( P s ( ω + ω ) + P s ( ω − ω ))Both expressions have a constant term independent ofthe signal—usually neglected since it is fourth-order inthe noise—and both have a term that decreases linearlywith the total measurement time. To compare thesetwo expressions, we note that for a uniform comb themeasurement time of the tunable LO must be N c timessmaller to account for the fact that the comb measure-ment is multiplexed. In addition, the power per combtooth for the comb version should be N c times smallerthan the tunable LO’s power if the mixer is to be opti-mally pumped. When there are M overlapping lines inthe IF, the signal-to-noise ratio (SNR) for both the comband the tunable LO are respectively given bySNR = 1 M N T N t N c σ | E | P s ( ω ( c ) n + ω )SNR = N T N t N c σ | E | P s ( ω + ω )In the best case scenario, where no lines overlap, theSNRs are identical. In the worst case scenario (a broad-band light source), M = N c and the SNR is a factor of √ N c times worse.In addition to additive noise, there is multiplicativenoise that arises from the fact that even incoherent sig-nals can have transient frequency domain correlations.Even when (cid:104) E s ( ω ) E ∗ s ( ω + nω r ) (cid:105) = 0, it will not be thecase that (cid:68) | E s ( ω ) E ∗ s ( ω + nω r ) | (cid:69) = 0. This manifestsas noise, and it effectively limits the dynamic range ofthe measurement. While the exact form depends onthe details of the source—white frequency noise differsfrom flicker noise, for example—the variance is typicallyproportional to equation (B1) for sufficiently broadbandspectra. Appendix C: Phase correction
In the derivation of this approach we assumed thatthe comb lines being used to probe the signal were freeof phase noise. In fact, this assumption can be relaxedif an additional dual comb spectroscopy measurement isperformed to measure their mutual phase fluctuationsand compensate for them. Suppose that the dual combbeating of pair n has been digitized and is given by V n ( t ) = E ( c ) n E ( c ) ∗ n e i ∆ n t , and that its magnitude hasbeen divided out to construct p n ( t ) = e i ( φ n − φ n ) e i ∆ n t .The correlation function is then given by C n ( ω ) = F T [ F ( ω − ∆ n , T ) F ∗ ( ω, T )]( − ∆ n )= 1 N T N t (cid:88) i,j e i ∆ n ( T j + t i ) e − iωt i S ( t i , T j ) F ∗ ( ω, T j ) . Since the total time is given by t = t i + T j , the explicitdependence on ∆ n can be removed by substituting in p n e i ( φ n − φ n ) and noting that the summation over i ismerely an FFT over the t i axis: C n ( ω ) = 1 N T N t (cid:88) i,j p n e i ( φ n − φ n ) e − iωt i S ( t i , T j ) F ∗ ( ω, T j )= e i ( φ n − φ n ) (cid:104)F t [ p n S ]( ω, T ) F ∗ ( ω, T ) (cid:105) T This expression is convenient since it eliminates any ex-plicit references to ∆ n by premultiplying the signal be-fore the spectrogram calculation. Not only does it removephase noise, but it also makes calculation of the powerspectrum more convenient, as it removes the global phaseof the dual comb lines. By defining the alternative cor-relation function ˆ C n ( ω ) asˆ C n ( ω ) ≡ (cid:104)F t [ p n S ]( ω, T ) F ∗ ( ω, T ) (cid:105) T (C1)we find that (cid:68) ˆ C n ( ω ) (cid:69) = (cid:12)(cid:12)(cid:12) E ( c ) n E ( c ) n (cid:12)(cid:12)(cid:12) P s ( ω ( c ) n + ω ) . This result guarantees that phase noise does not con-tribute to the reconstruction, essentially by eliminatingthe phase entirely. Figure 5 illustrates this for the datain Figure 2 for free-running combs. The combs have ran-dom walk phase noise producing offset fluctuations witha 1 MHz FWHM and repetition rate fluctuations with a1 kHz FWHM, similar to what is found in free-runningQCLs. As a result, the reconstruction produced by theprevious approach fails. However, using the alternativecorrelation defined in (C1), the correct result is recov-ered.
Appendix D: Detector nonlinearity
Because we are using many frequencies to reconstructour signal, it is possible that detector/mixer nonlinearity -5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.6-5 -4 -3 -2 -1 0 1 2 3 4 500.51-5 -4 -3 -2 -1 0 1 2 3 4 5012-5 -4 -3 -2 -1 0 1 2 3 4 5
Offset frequency (GHz)
Phase noise correction actualuncorrectedOffset from4220 GHz Offset from4560 GHz Offset from4780 GHz Offset from4800 GHz P o w e r s p e c t r a l d e n s i t y ( n W / M H z ) corrected Figure 5. Spectral reconstruction in the presence of phasenoise. Offset fluctuations of 1 MHz and repetition rate fluc-tuations of 1 kHz preclude reconstruction, but using the al-ternative correlation function correctly recovers the originalspectra. could negatively impact the results of the measurement,for example by distorting the signal. For example, thiscan be a challenge in dual comb spectroscopy when highlynonlinear mixers such as hot electron bolometers are usedto measure the spectrum [6], as nonlinearity will causedifferent comb lines to mix. However, in this case we donot expect such effects to be significant. When operatingin the heterodyne limit, the measured signal is given by S i ( t ) = 12 | E c i ( t ) + E s ( t ) | = 12 | E c i | + E ∗ c i E s + 12 | E s | and the | E c i | term is much larger than the heterodyneterm. Even so, it is easy to ignore because it only beatsat multiples of the repetition rate (i.e., is periodic). Anynonlinearity will act upon it only to produce another sig-nal that beats at the repetition rate, and we can thereforeexpect to continue to be able to ignore it.We therefore expect that the lone effect of nonlinearityon the heterodyne measurement is to reduce its sensitiv-ity. For example, if a standard two-level saturation modelis used to model the detector response, then the outputsignal will be related to the input power by S ( t ) = P ( t )1 + P ( t ) P sat where P sat is the saturation power. The heterodyne re-sponsivity is essentially the differential response of thismodel, which is squared since both detectors suffer thisdecrease: (cid:18) dSdP (cid:19) = (cid:18) PP sat (cid:19) − -0.0200.02-0.0200.02-0.0200.02-0.0200.02 -5 -4 -3 -2 -1 0 1 2 3 4 500.020.04-5 -4 -3 -2 -1 0 1 2 3 4 500.050.1-5 -4 -3 -2 -1 0 1 2 3 4 500.050.10.15-5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency (GHz)Offset from nearest comb 1 line
/P sat absoluteresponsivityheterodyneresponsivityactual spectrum (x 1/16)reconstructed spectrumOffset from4220 GHz Offset from4560 GHz Offset from4780 GHz Offset from4800 GHz P o w e r s p e c t r a l d e n s i t y ( n W / M H z ) R e s i d u a l ( n W / M H z ) Effect of mixer nonlinearity on spectrum reconstruction
Figure 6. Spectral reconstruction in the presence of mixernonlinearity. Although the reconstruction is a factor of 16times lower due to a reduction in the heterodyne sensitivity,it is otherwise unaffected.
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