Spatial multiplexing of soliton microcombs
Erwan Lucas, Grigori Lihachev, Romain Bouchand, Nikolay G. Pavlov, Arslan S. Raja, Maxim Karpov, Michael L. Gorodetsky, Tobias J. Kippenberg
SSpatial multiplexing of soliton microcombs
E. Lucas, G. Lihachev,
2, 3
R. Bouchand, N. G. Pavlov,
2, 4
A. S. Raja, M. Karpov, M. L. Gorodetsky,
2, 3 and T. J. Kippenberg ∗ IPHYS, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Russian Quantum Centre, 143025, Skolkovo, Russia Faculty of Physics, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia
Dual-comb interferometry utilizes two optical fre-quency combs to map the optical field’s spectrumto a radio-frequency signal without using mov-ing parts, allowing improved speed and accuracy.However, the method is compounded by the com-plexity and demanding stability associated withoperating multiple laser frequency combs. Toovercome these challenges, we demonstrate simul-taneous generation of multiple frequency combsfrom a single optical microresonator and a singlecontinuous-wave laser. Similar to space-divisionmultiplexing, we generate several dissipative Kerrsoliton states – circulating solitonic pulses drivenby a continuous-wave laser – in different spa-tial (or polarization) modes of a MgF microres-onator. Up to three distinct combs are producedsimultaneously, featuring excellent mutual coher-ence and substantial repetition rate differences,useful for fast acquisition and efficient rejectionof soliton intermodulation products. Dual-combspectroscopy with amplitude and phase retrieval,as well as optical sampling of a breathing soliton,is realised with the free-running system. Compat-ibility with photonic-integrated resonators couldenable the deployment of dual- and triple-comb-based methods to applications where they re-mained impractical with current technology. Introduction
Shortly after the inception of the optical frequencycomb , it was realised that combining two combs withslightly different repetition rates on a photodetector pro-duces a radio-frequency (RF) interferogram that sam-ples the optical response , without any moving parts.Such dual-comb techniques have been demonstrated inboth real-time and mid-infrared spectroscopy, dis-tance measurements , two-way time transfer , coherentanti-Stokes Raman spectro-imaging , as well as photonicanalogue to digital conversion . However, facing thecomplexity and cost associated with operating two laserfrequency combs, novel methods are being actively ex-plored with a view to reduce the system complexity andinherently improve the mutual coherence. For example,instead of phase locking two independent conventional ∗ tobias.kippenberg@epfl.ch mode-locked lasers, both combs can be generated in thesame laser cavity , via repetition rate switching of a sin-gle comb , or spectrally broadened in the same fibre inopposite propagation directions . As the noise sourcesare common mode, the relative coherence between thecombs is significantly improved, allowing for longer co-herent averaging .Recent advances in the field of high-quality-factor mi-croresonators pumped with a continuous wave (CW)laser have led to the discovery of ‘Kerr’ frequencycombs (also termed ‘microcombs’) that arise due tononlinear wave mixing mediated by the optical Kerreffect. One particular state of such combs corre-sponds to the formation of dissipative Kerr solitons (DKSs) — self-localised pulses of light circulating inthe resonators arising from the double balance be-tween loss and parametric gain and between disper-sion and nonlinearity . The coherent and broad-band properties of DKS-based microcombs have alreadyfound applications in parallel coherent communication ,distance measurement , astrophysical spectrome-ter calibration , self-referencing , and photonic-integrated frequency synthesis . DKSs display rich non-linear dynamics, such as dispersive wave emission , Ra-man self-shifting , or breathing solitons .The initial demonstrations of dual-microcomb appli-cations relied on pairs of physically distinct yet almostidentical resonators . Recent works demon-strated the generation of dual-DKS combs with counter-propagating solitons within the same spatial mode of asingle microresonator, using the clockwise and counter-clockwise mode degeneracy, and showed a drastic im-provement of the coherence. However, this technique islimited to counter-propagating pumps and as such re-quires nonreciprocal elements, i.e. circulators. Moreover,since the same mode family is used, only small relativecombs offsets are possible, while the repetition rate dif-ference is induced via the Kerr and Raman effects andremains relatively moderate. As a result, the correspond-ing RF comb is not centred at sufficiently high frequen-cies, and RF lines near DC may overlap, which can leadto soliton-locking , but also implies that several pairsof lines beat at identical RF frequencies. Likewise, thesmall repetition rate difference restricts the acquisitionspeed. Finally, the scheme is inherently limited by thetwofold degeneracy of whispering gallery modes (WGM),only allowing dual-comb generation.Like optical fibres, optical microresonators can also ex- a r X i v : . [ phy s i c s . op ti c s ] S e p PumpComb lines (cid:47)(cid:374)(cid:410)(cid:286)(cid:396)(cid:373)(cid:381)(cid:282)(cid:437)(cid:367)(cid:258)(cid:415)(cid:381)(cid:374)
Sideband (cid:75)(cid:393)(cid:415)(cid:272)(cid:258)(cid:367)(cid:3)(cid:296)(cid:396)(cid:286)(cid:395)(cid:437)(cid:286)(cid:374)(cid:272)(cid:455)(cid:68)(cid:381)(cid:282)(cid:286)(cid:3)(cid:296)(cid:258)(cid:373)(cid:349)(cid:367)(cid:455)(cid:3)(cid:1005)(cid:68)(cid:381)(cid:282)(cid:286)(cid:3)(cid:296)(cid:258)(cid:373)(cid:349)(cid:367)(cid:455)(cid:3)(cid:1006)
ResonatorSSB modulatorCWPump Laser EDFA N o t c h OSAESAPD
10 µm E M e n e r g y d e n s . ( a r b . un . ) Co-propagating
Pump + Sideband
Counter-propagating
Sideband PumpPump + Sideband Sideband
Triple comb
Sidebands Pump eda cb
Figure 1.
Principle of spatial multiplexing of solitons in a single microresonator (a)
Crystalline MgF whisperinggallery mode (WGM) resonator used in this work. (b) Simulation of several optical mode profiles supported by the WGMprotrusion. (c)
Schematic representation of the three schemes applied. (d)
Setup for dual DKS generation via spatial multi-plexing in the co-propagating direction. The single-sideband (SSB) modulator creates an additional carrier to pump a secondmode family. EDFA: erbium doped fibre amplifier. E/O-SA: electronic/optical spectrum analyser. (e)
Principle of multiplexedcombs generation. The main pump laser (red arrow) is modulated to generate one sideband (blue arrow). The laser andsideband pump one resonance of two different mode families (1 in red and 2 in blue) and generate a soliton comb in each ofthem through the Kerr effect (red and blue lines). Co-propagating pulses may experience intermodulation effects (see SI fordetails). hibit multiple spatial mode families, which provide addi-tional degrees of freedom in which light can propagate.In fibre optical communication, space-division multiplex-ing utilizes different spatial modes of an optical fiber asadditional parallel channels to transmit data . It re-mains an open question if dual-comb can be generated inan analogous way within different spatial modes of a cav-ity. The presence of higher-order modes is known to im-pact DKSs, causing alterations of the comb envelope viaavoided mode crossing , dispersive wave emission ,repetition rate instability , intermode breathing ,and Stokes soliton generation . Although pumping oftwo orthogonally polarised modes was investigated inpreliminary works , generating independent solitonstates in distinct spatial modes was not shown to date. Here, we demonstrate spatial multiplexing of DKSstates in a microresonator. We simultaneously generatemultiple DKS-based combs within distinct spatial modesof a single optical microresonator. Up to three differentmode families of the same polarisation are pumped usinga laser and modulation sidebands (Fig. 1c). This spatialmultiplexing thereby allows not only dual but also triplefrequency comb generation from one and the same device,which to the best of the authors knowledge, has not beenachieved with any other laser frequency comb platformto date (e.g. Ti:Sa, semiconductor or fibre-based modelocked lasers). The technique introduced here also over-comes several of the previous shortcomings. The distinctfree spectral ranges of the respective mode families en-able the generation of independent soliton pulse streams O p t i c a l po w e r d B / d i v . Span 1 nm d B / d i v -100 -50 0 50 100 150MHz + 4.282 GHz R F po w e r d B / d i v . CF 12.4078 GHz d B / d i v .
655 kHz -20 -10 0 10 20kHz + 4.288 GHz R F po w e r d B / d i v . RBW 100 Hz Time (1 µs/div) V o l t age ( a r b . un i t ) Time (10 ns/div) V o l t age ( a r b . un i t ) ab cd e Multipexing(RBW 100 Hz) Distinctresonators(RBW 1kHz)
Figure 2.
Dual-comb generation with spatially-multiplexed co-propagating solitons (a)
Generated dual-comb opticalspectrum. The DKS-based combs are interleaved and spaced by ∼ GHz (see inset). The red markers delineate one combfrom the other. (b)
Resulting dual-comb RF heterodyne beatnotes. Resolution bandwidth (RBW): 3 kHz. The line spacing(repetition rate difference shown in inset) is 655 kHz. CF: centre frequency. (c)
Focus of one line of the RF comb. Theblue trace denotes the multiplexed solitons in (b). The red represents the results from solitons generated in two distinctmicroresonators pumped with a single laser. (d)
Temporal interferogram of the dual-comb heterodyne shown in (b), recordedon a fast sampling oscilloscope, and after digital bandpass filtering to select the RF comb. (e)
Detail of the temporal trace (d)when the two pulses overlap ( ∼ × magnification). with substantial repetition rate differences (100 kHz –100 MHz). As a single laser and resonator are used,the resulting combs have thus excellent mutual coher-ence, and support dual-comb spectroscopy (with ampli-tude and phase retrieval) in spite of using a free runningsystem. The larger offset between combs prevents soli-ton locking and associated mapping ambiguity. Themultiplexing can also be performed in co- or counter-propagating directions. Finally, the scalability is demon-strated with the generation of three combs in a singleresonator. Beyond established dual-comb techniques ,triple comb can be used for higher dimensional spec-troscopy, with the potential to increase information con-tent, accuracy or speed of acquisition, such as in 2D co-herent spectroscopy , as well as advanced comb-baseddistance measurement with increased ambiguity range .The multiplexing approach could lead to significant sim-plifications in the implementation of these schemes. Results
We take advantage of the multi-mode nature of a crys-talline MgF WGM cavity (Fig. 1a). The fabrication ofMgF crystalline cavities by diamond turning and subse-quent polishing with diamond slurries leads typically tomultimode resonators with several mode families reach-ing ultrahigh quality factors ( Q ) exceeding . In thepresent work, the resonator used has a free spectral range(FSR) of 12.4 GHz, and features up to 5 mode familieswith the same polarisation that sustain DKS formation ( Q ≥ ), as shown in Fig. S.1 of the supplementaryinformation (SI). The light is coupled using a taperedoptical fibre, whose position is adjusted to tune the rela-tive coupling rate to higher order modes. Spatial multiplexing with co-propagating pumpfields — We first study the use of co-propagatingpumps. In this scheme (Fig. 1d,e), simultaneous pump-ing of two soliton-supporting resonances is achieved viaelectro-optical modulation. The light of an initial pumplaser (external cavity diode laser, wavelength 1554 nm)passes through an IQ-modulator to generate a singlesideband , without fully suppressing the carrier, suchthat both reach the same power level. This createstwo mutually phase-coherent carriers with a tunablefrequency offset. The modulation frequency is set to f m = ω m / π ∼ . GHz to match the separation oftwo soliton-supporting resonances, which belong to dif-ferent spatial mode families but have the same polari-sation. The ‘laser scanning technique’ is subsequentlyapplied on the main pump laser to trigger DKS forma-tion in both mode families simultaneously. A successfultuning is however challenging as each resonance inducesa thermal shift. This is mitigated in two ways: the laseris tuned across the resonances using the diode current,which allows tuning speeds faster than the thermal re-laxation time of the cavity (typically ms timescale). Sec-ond, the modulation frequency f m is carefully adjustedso that the ‘high detuning end’ of both soliton steps approximately coincide (Fig. S.1b-c). We observed that O p t i c a l po w e r d B / d i v . CF 12.4075 GHz d B / d i v . -100 -50 0 50MHz + 2.74899 GHz R F po w e r d B / d i v . CF 2.76161 GHzSpan 20 kHz R F po w e r d B / d i v . RBW:200 Hz CF 2.78682 GHzSpan 20 kHzRBW:200 Hz Time (2 µs/div) V o l t age ( a r b . un i t ) A m p li t ude
191 192 193 194 195Optical frequency [THz]-505 P ha s e (r ad ) Data Programmed
CWPump Laser EDFA EDFAEDFAEDFA ESASCOPDWaveshaper10%90% CCWcombCWcomb refsignal
OSAOSA a efb cd
100 GHz ref.signal
Figure 3.
Dual-comb generation with spatially multiplexed counter-propagating solitons and proof-of-principlespectroscopy (a)
Setup for counter-propagating dual and triple DKS-comb generation and spectroscopy. (b)
Optical spectraof the two counter-propagating combs. The inset shows the two repetition rate beats of the combs (CF: centre frequency). (c)
Resulting dual-comb beatnote (RBW 3 kHz). (d)
High resolution spectra of two lines of the RF comb in (c). (e)
Temporalinterferogram of the dual-comb heterodyne for the signal path (with waveshaper) and reference path. (f )
Retrieved amplitudeand phase of the signal interferogram produced by coupling the dual-soliton pulse trains through a waveshaper programmedwith synthetic absorption features (100 GHz FWHM). The orange lines display the programmed functions. this maximises the success rate of simultaneous singleDKS initiation in both mode families (see SI for details).After generation, the main laser is locked to the microres-onator via offset Pound-Drever-Hall (PDH) locking andthe dual-DKS combs can be stably maintained for morethan 12 hours.In this manner two simultaneous streams of DKSs areproduced. The optical spectrum of the microresonatoroutput (Fig. 2a) shows the two interleaved DKS combsoffset by f m . The repetition rates of the two combs dif-fer by ∆ f rep = 655 kHz (around 12.4 GHz). This corre-sponds to a relatively small spectral compression factor of m = f rep / ∆ f rep = 1 . × , which is useful to in-crease the acquisition speed of a moderate optical span.The beating of the dual-comb results here in an RF combcentred at f m = 4 . GHz. The individual lines of the RFcomb are still resolution-limited at 100 Hz bandwidth,although the system is free running (neither the combs’repetition rates nor the pump wavelength are stabilised).In comparison, two combs generated in distinct microres-onators pumped with the same laser have a linewidthbroader than 1 kHz. Thanks to the high centre frequency f m , the total ∼ MHz span of the RF comb can bemapped into the corresponding 3 THz of optical spanwithout aliasing at baseband frequencies and no signs ofinter-soliton locking were observed consequently. Nev-ertheless, it may seem that a high centre frequency is a drawback, as it requires the use of a very fast samplingrate to directly acquire the dual comb interferogram, butsince this frequency is set by the modulation f m , it is pos-sible to downmix the RF comb to baseband with an IQdemodulator and relax the requirement on the samplingrate.Importantly, although the solitons circulate in dis-tinct spatial modes, they can interact via four-wave mix-ing (FWM) when co-propagating and effectively becomemodulated at the rate at which the solitons cross ∆ f rep ,as detailed in the supplementary information (SI, cf.Fig. S.3). The modulation products that arise aroundthe comb lines will beat with adjacent comb lines at fre-quencies identical to the RF comb and may thus induceoptical-to-RF mapping ambiguities. In the present case,the relative strength of the first intermodulation sidebandis approximately − dBc (see Fig. S.3e), in agreementwith the effect of cavity filtering. This value is sufficientlyweak to be neglected in most practical applications.Another pair of mode families can be selected toachieve a larger repetition rate difference (see Fig. S.2, ∆ f rep = 9 . MHz) when a faster acquisition is targeted.The compression m = 1 . × is more than one or-der of magnitude lower than the previous demonstra-tions with counter-propagating solitons ( . × in and . × in ), whilst typical mode-locked lasersare in the range × − . Note that increasing d B / d i v . d B / d i v . t = 0.00 µst = 0.20 µs-0.5 0 0.5Frequency - 7.5 GHz (GHz) t = 0.40 µs A m p li t ude ( a r b . un i t ) F a s t t i m e [ p s ] F a s t T i m e ( p s ) p s / d i v . EDFA SCOPD
PumpSideband t Stable ref. pulses t Breathing soliton a b dce f
Direct sampling Dual comb optical sampling
Figure 4.
Resolving the breathing dynamics of a soliton (a)
Experimental setup for breathing DKS dual comb imaging. (b)
Optical spectrum of the breathing soliton pulse train and (c) of the stable reference pulse train. (d)
RF spectrogramof the breathing soliton interferogram taken at maximum spectral contraction ( t = 0 µ s ) and expansion ( t = 0 . µ s ). Thearrow marks the pump position. (e) Spatiotemporal imaging of a breathing DKS via direct real-time sampling of the pulsetrain. (f )
Same measurement realised with the multiplexed dual-comb. The fast time resolution is improved by an order ofmagnitude. The inset magnifies the white rectangular window. ∆ f rep enables even stronger intermodulation suppression( ∼ − dBc here) as the sidebands are created well out-side the cavity bandwidth (see SI, Fig. S.3f). These ex-periments illustrate the flexibility of the technique andits potential to substantially increase the bandwidth ofthe dual-comb interferogram and acquisition speed, com-pared with prior schemes using counter-propagating soli-tons . Although a collinear dual-comb is not suitablefor some applications, co-propagating soliton generationsimplifies the scheme considerably, as it lifts the need fornonreciprocal devices. Furthermore, two combs gener-ated this way could be separated via de-multiplexing, ifthe offset f m is high enough (e.g. beyond 25 GHz), whichshould be possible for integrated micro-resonators withlarger FSR ( > GHz). Alternatively, if the pumpedmodes are orthogonally polarized, they could be demul-tiplexed with a simple polarisation beam splitter.
Spatial multiplexing in the counter-propagatingconfiguration — Alternatively, the two spatial modefamilies can be excited in a counter-propagating way,analogous to previous implementations . First, thepump laser is split unevenly between two paths. A pairof circulators is then used to couple light into the res-onator and to collect the transmitted combs on both sides(Fig. 3). 90% of the amplified pump power ( ∼ mW)is coupled directly into the counter-clockwise (CCW) di-rection. In the other path, the remaining 10% of thepump is sent through a single sideband modulator oper-ated in carrier-suppressing mode to frequency-shift the light by the offset separating the two resonances. Afteramplification to a similar power level of ∼ mW, thefrequency-shifted light is coupled in the clockwise (CW)direction.We used another set of mode families whose resonanceoffset is f m = 2 . GHz, and the repetition rate differenceis ∆ f rep = 371 kHz to demonstrate the counter-clockwisespatial multiplexing of DKSs. The soliton formation istriggered in the same way as in the co-propagating case.The two generated single-soliton combs are shown inFig. 3. Each comb has an average power of ∼ µ W atthe output of the resonator (after excluding the pump).The corresponding RF comb (Fig. 3) features a sim-ilar degree of stability to the co-propagating scheme,with 200 Hz wide beatnotes throughout the RF comb(Fig. 3d). A main advantage of the counter-propagatingpump configuration, is the absence of intermodulationproducts in the combs (Fig. S.3g). This is to be expected,as in this case, the FWM process between two differentcombs is momentum-forbidden (see SI for details).With this pumping configuration the combs can also beaccessed individually, allowing the implementation of awider range of dual-comb applications. As a proof of con-cept spectroscopy experiment, the combs are amplified to ∼ mW average power and one comb is sent througha waveshaper before interfering with the second comb.The beating is recorded on a high sampling rate oscil-loscope (1 ms acquisition time, corresponding to ∼ averages). The amplitude and phase of the RF comb O p t i c a l po w e r d B / d i v . CF 12.4073 GHz d B / d i v .
380 kHz734.9 kHz 355kHz -50 0 50MHz + 2.17139 GHz R F po w e r d B / d i v . -100 -50 0 50 100MHz + 2.42351 GHz R F po w e r d B / d i v . -40 -30 -20 -10 0 10 20 30 40MHz + 4.5949 GHz R F po w e r d B / d i v . a bcde Comb 2Comb 1 Comb 1 + 2Comb 2 + 3Comb 1 + 3 Co m b Combs generator (cid:24)(cid:258)(cid:410)(cid:258)(cid:3)(cid:258)(cid:272)(cid:395)(cid:437)(cid:349)(cid:400)(cid:349)(cid:415)(cid:381)(cid:374)& processingSampleLO Photon echoPumpProbe + emission (cid:258) (cid:271) (cid:400) (cid:381) (cid:396) (cid:393) (cid:415) (cid:381) (cid:374)
Figure 5.
Triple comb generation in a single resonator by multiplexing in three mode families (a)
Optical spectra.Comb 1 is generated in the CCW direction, while Comb 2 and 3 are generated in the CW direction. The inset shows the threedistinct repetition rate beats. (b-d)
Heterodyning the three pulse trains on the same photodiode leads to the formation ofthree RF combs (RBW 3 kHz). (e)
Envisioned application of the triple-comb generator for two-dimensional spectroscopy teeth are compared to a reference signal recorded with-out the waveshaper. Figure 3 shows that the retrievedamplitude and phase closely match the programmed syn-thetic resonance profiles over a span of 4 THz.Rapid coherent linear optical sampling was also re-alised to resolve the dynamics of a DKS pulse breath-ing at a rate ∼ MHz. Indeed, the fast recordingof the interferogram between a DKS comb and a ref-erence comb offers the possibility to spectrally resolvethe soliton dynamics in the microresonator, as recentlydemonstrated using an electro-optic comb as reference.Here, the multiplexing scheme in counter-propagation isapplied instead and the detuning in each direction is care-fully set in order to generate a CCW breathing pulse anda stable DKS in the CW direction that serves as a ref-erence (Fig. 4a). Another pair of mode families yieldinga higher repetition rate difference ( ∆ f rep = 9 . MHz)is selected, such that the acquisition speed of the inter-ferogram is faster than the breathing rate. The opticalspectra of each pulse train can be viewed in panels band c. The breathing soliton spectrum features a typ-ical triangular profile on the optical spectrum analyserdue to the averaging of the periodic spectral broadeningand compression. The real-time spectral evolution of thebreathing soliton can be retrieved by taking the Fouriertransform to the dual comb interferogram (Fig. 4d, seeMethods for details). The salient features of breathingDKS can be retrieved : over half a breathing period, thespectrum contracts and expands. Furthermore, the comblines located near the pump are oscillating out of phasefrom the wing. The detection of the dual comb interfer- ogram envelope allows the spatiotemporal dynamics ofthe breathing soliton to be mapped. After accounting forthe compression ratio of the dual comb acquisition, thismethods yields an effective temporal resolution of ∼ ps(Fig. 4f) and represents a 10 fold improvement comparedto the direct real-time sampling method (Fig. 4e, see SIfor details). Triple soliton comb generation via spatial multi-plexing — We next demonstrate the ability to multi-plex three soliton combs by pumping three mode fami-lies simultaneously. We employ the counter-propagatingconfiguration, but combine two tones on the modula-tor (Fig. 3): f m = ω m / π = 2 . GHz and f (cid:48) m = ω (cid:48) m / π = 4 . GHz. This allows two mode families tobe co-pumped, and thus the creation of two combs inthe CW direction, while another comb is generated bypumping a third mode family in the CCW direction. Re-markably, the excitation technique outlined earlier, wasalso applied successfully to generate all three single soli-ton state combs (Fig. 5a,b). Heterodyning the combscreates a set of three RF combs centred at f m , f (cid:48) m and | f (cid:48) m − f m | = 2 . GHz and with a line spacing of 380 kHz,355 kHz, and 735 kHz respectively. In the heterodyne RFcombs between the CCW and each of the CW combs, weare able to observe weak additional lines resulting fromthe intermodulation products on each of the CW solitoncombs (Fig. 5b,d). The spikes around ± MHz on theseRF-combs are caused by the PDH phase modulation. Im-portantly, these additional beatnote products give rise tospectrally distinct frequencies and can thus be removedduring signal processing (by only selecting the frequencycomponents at e.g. f m + n ∆ f rep ), and critically do notinduce mapping ambiguities.The triple soliton comb configuration with two co-propagating combs could find applications in advancedspectroscopy schemes such as two-dimensional spec-troscopy . A recent demonstration employed threeTi:Sa lasers, two of which generated a pump and probepulse trains to excite a photon echo in Rubidium vapour,which was heterodyned with the third local oscillatorcomb allowing the fast acquisition of 2D spectra witha single photodiode. The multiplexing approach wouldoffer a major simplification and cost reduction of suchschemes, as illustrated in Fig. 3e. Optical distance mea-surements can also benefit from this triple comb scheme,as sending a pair of combs onto the target would providetwo series of synthetic wavelength chains, allowing a greatextension of the ambiguity range without compromisingthe resolution. Such scheme was recently demonstratedusing three electro-optic combs reaching an accuracy of750 nm over 80 m distance (instead of 15 mm with thedual comb method). Discussion
In summary, spatial multiplexing of soliton combs in asingle microresonator is demonstrated experimentally forthe first time, both in co- and counter-propagating pumpconfiguration. The multiple soliton pulse streams haveexcellent mutual coherence, and their frequency offset is asubstantial fraction of the repetition rate, which preventsmapping to negative frequencies. The generated dual-combs are shown to be suitable for spectroscopy. Largerelative differences in repetition rates can be obtained,enabling fast acquisition and improved bandwidth usage.Using the birefringence of MgF even larger differencesare possible, when pumping modes with orthogonal po-larisations ( > MHz were observed, as demonstratedin the SI). When combined with faster repetition rates,such configurations could find applications in RF sig-nal processing and acquisition , or for ultra-rapid vi-brational spectroscopy in condensed matter. The fastrecording of a dual DKS-comb heterodyne also prove use-ful to investigate soliton dynamics with unprecedentedresolution, such as measuring the line-by-line spectraldynamics of a breathing soliton. We also demonstratethat a high repetition rate difference is also beneficial tosuppress the intermodulation products when the solitonsare co-propagating (as shown in Fig. S.3c).The presented scheme is already within reach of micro-fabricated ring waveguide resonators , as illustratedby the recent demonstration of a device supporting soli-tons in both the TE00 and TM00 mode families , fortwo closely spaced pump frequencies. In the future, thisimproved fabrication control will allow the control ofthe mode frequency separation and repetition rate dif-ference, while mitigating the impact of modal crossing.Furthermore, waveguide geometric dispersion engineer-ing will enable larger bandwidth coverage and cen-tral wavelength selectivity . The simplicity of the co- propagating scheme makes it compatible with full on-chipintegration, as all the elements are readily available inphotonic integrated circuits.The method is flexible and easily scalable, as shown bythe generation of three simultaneous soliton combs – sofar out of reach for other frequency comb platforms. Thismultiple comb source has the potential to extend the ca-pabilities of comb-based method, for greater informationcontent, accuracy or speed of acquisition. Methods
Resonator fabrication and characteristics — Themicroresonator protrusion was fabricated via high-precision diamond turning of a mono-crystalline MgF blank followed by hand polishing with diamond slurriesand cleaning. The FSR of 12.4 GHz corresponds to amajor radius of 2.8 mm. The resulting WGM protru-sion can be approximated by an oblate spheroid with awidth of µ m and a height of µ m (this very shallowprotrusion appears almost flat in the picture Fig. 1a).While single mode cavity protrusions have been demon-strated using advanced micro-machining techniques, wetarget a wide protrusion instead, which supports a largenumber of WGM modes and makes the polishing lesschallenging. The obtained quality factors of the soliton-supporting resonances are above at 1554 nm. Thegroup velocity dispersion (GVD) of MgF is naturallyanomalous in the C-band ( β ∼ − . / mm ), so thatno geometric dispersion engineering is needed to reachthis dispersion regime, which is necessary for soliton for-mation. Due to the loose confinement and the large mainradius of the structure, higher order modes have essen-tially a higher FSR without significantly changing thedispersion which is dominated by the material. Over-all, 5 resonators were fabricated, featuring at least twosoliton-supporting mode families (FSR 8, 12, 14, 17 and26 GHz).Evanescent coupling to the WGM is achieved with a ta-pered optical fibre. The mode of the tapered fibre and ofthe WGM resonator are not orthogonal to each other andone taper mode can excite several higher order WGM, al-though the mode overlap and phase matching conditionare different for each WGM thus leading to variationsof their respective coupling rate. However, shifting theoptical fibre position out of the equatorial plane of theresonator, influences the coupling strength of the evenand odd WGMs (p. 59) (in the polar direction). Thisdegree of freedom is used in the experiment to adjust thecoupled power in the modes producing the solitons. Dual comb imaging of the soliton dynamics —In a specific range of detuning and pump parameters,DKS can undergo a breathing instability where the soli-tonic pulse oscillates both in amplitude and duration .Thanks to the long photon lifetime of microresonators,the oscillation period is much longer than the roundtriptime (the breathing rate is ∼ MHz, for a 12.7 GHzrepetition rate).First, resolving the spatiotemporal dynamics of breath-ing solitons was carried out via direct sampling of thebreathing pulse train, using a very fast photodiode( ∼ ps impulse response) and real-time oscilloscope(120 GSa/s) in order to sample every roundtrip . How-ever, the fast temporal resolution of this method is lim-ited by the photodiode impulse response and closelyspaced solitons may not be distinguished (Fig. 4e), whilethe slow breathing evolution is oversampled and can onlybe monitored for a short time span at such samplingrates.Another method is to take advantage of the dual combprinciple. This method is derived from coherent linearoptical sampling . The envelope of a dual comb in-terferogram between a breathing soliton and a referencepulse train with a slight difference in repetition rate ∆ f rep yields the convolution of the breathing pulse with thereference. The pulses overlap each / ∆ f rep , which corre-sponds to the time for one pulse to sweep over one entireroundtrip of the other pulse train, and sets the imagingframe rate. Thus if ∆ f rep is faster than the breathingrate, the breathing dynamics can be monitored, while re-laxing the requirement on the sampling rate (ultimatelyonly to match RF comb bandwidth).The dual comb is generated via multiplexing two soli-tons in counter-propagation (see Fig. 4). The mode fam-ilies are selected to reach ∆ f rep = 9 . MHz (one frameperiod is acquired over 1340 roundtrips). The real-timespectral evolution of the breathing soliton (multiplied bythe reference pulse spectrum) can be retrieved by tak-ing the Fourier transform of the dual comb interferogram(Fig. 4d). In order to improve the signal to noise ratio,multiple interferogram frames at similar breathing phaseswere averaged together (after multiplication by a gaus-sian window of width / ∆ f rep ).To view the spatiotemporal dynamics of the breath-ing soliton, the interferogram envelope is retrieved viaHilbert transform and each frame is sliced and stacked (Fig. 4f). The fast time axis can be rescaled to span /f rep to account for the compression ratio of the dualcomb acquisition method. Panels e and f show the vastimprovement in temporal resolution of the dual combmethod over the direct sampling method ( ∼ ps vs. ∼ ps). Data availability statement
The data and code used to produce the results of thismanuscript will be available on Zenodo upon publication.
Authors contributions
E.L. and G.L. designed the experimental setup. E.L.performed the experiments and analysed the data.G.L. fabricated the device, with assistance of N.G.P..E.L., R.B. and A.R. performed the experimental comblinewidth measurement. M.K. and A.R. assembled theRF components for the single sideband modulator driv-ing. E.L. wrote the manuscript, with input from otherauthors. T.J.K. and M.L.G. supervised the project.
Acknowledgments
The authors thank Dr. Nathan Newbury for impor-tant suggestions and comments. The authors thank J.D.Jost and W. Weng for their assistance as well as J. Liu,H. Guo, N.J. Engelsen and M. Anderson for their feed-back on the manuscript. This publication was supportedby funding from the Swiss National Science Foundationunder grant agreement 163864, by the Air Force Office ofScientific Research, Air Force Material Command, USAFunder Award No. FA9550-15-1-0099, and by the Ministryof Education and Science of the Russian Federation un-der project RFMEFI58516X0005. E.L. acknowledges thesupport of the European Space Technology Centre withESA Contract No.: 4000118777/16/NL/GM. Lomsadze, B., Smith, B. C. & Cundiff, S. T. Tri-combspectroscopy (2018). 1806.05071. Hansch, T. W. Nobel Lecture: Passion for precision.
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Identification of soliton resonances — The piezo ofthe pump ECDL is scanned over a full FSR of the cav-ity at 1554 nm, while recording the transmission and thenonlinearly generated light on an analogue to digital con-verter. Five resonances featuring the typical step transi-tion associated with the formation of solitons could beidentified (Fig. S.1a). Their relative offset was estimatedusing piezo voltage calibration. Tuning Method — In order to initiate the dual-combformation, a pair of target mode families is first identified.The pump laser is tuned close to one resonance and themodulation frequency of the sideband is set close to theresonance offset, while the bias of the SSB modulator isadjusted to generate a blue or red sideband, depending onthe sign of the frequency shift needed. After a coarse ad-justment, both resonances are visible when scanning thepump laser over a small frequency span (Fig. S.1b). Thecoupling can be optimized by changing the tapered fibreposition in order to increase the soliton step length inboth families. The final adjustment consists of tuning thesideband frequency shift so that the large detuning end ofeach the soliton step becomes aligned (Fig. S.1c). Whenscanning the laser, two soliton states can then be excitedsimultaneously more than 50 % of the time. Upon tuningthe laser across resonance, the number of generated soli-tons in one state is typically stochastic, due to the chaoticmodulation instability that seeds the solitons . However,the single soliton state is the most attractive state due toits smooth envelope, but remains challenging to obtain inmicroresonators. We found that our procedure also im-proves the success rate for dual single soliton generation.We believe this is because the tuning to the ‘high detun-ing end’ makes it more favourable for solitons to collide ordecay . Nonetheless the dual single soliton productionremains less probable, with an estimated success rate be-low 10 %. This rate could be improved by implementingan active capture feedback mechanism . Larger repetition rate difference via selection ofmode families — Several RF comb offset frequenciesand repetition rate differences can be achieved in thesame resonator, by changing the pair of modes support-ing the solitons. In this way we could generate solitons inthe co-propagating direction (Fig. S.2a) with an offset of4.9 GHz and a repetition rate difference of ∼ MHz. Theresulting RF comb (Fig. S.2b) spans more than 4 GHz.However, this high offset frequency f m combined with abroader comb implies that the RF comb extends beyond f rep / and thus overlaps with the mirror comb centredat f rep − f m , leading to potential mapping ambiguitiesin the overlap region. Engineering the modes of the mi-croresonator, enabled by better fabrication control, willallow an optimal bandwidth usage. Very large repetition rate difference via pumpingof orthogonally polarised modes — The microres-onator not only supports higher order spatial modesbut also fundamentally orthogonally polarized modes.
DualComb ab c
Spatial Mode 1 Spatial Mode 2
01 0.7 0.2 0.1 0.7Laser scan G en . L i gh t ( a r b . un i t ) Laser scan
Figure S.1. (a)
Identification of the soliton-supporting reso-nance over one cavity FSR. The graphs display the generatedcomb light at the output of the resonator as the laser fre-quency is decreased. The step features correspond to the de-tuning region where solitons exist. (b)
Sequential excitationof two soliton-supporting resonances (in the co-propagatingscheme), when the offset frequency is detuned. The left reso-nance is excited with the pump laser light while the right isexcited with the sideband. (c)
Adjusting the sideband shiftallows the overlap of the resonances. The region where thetwo steps coexist corresponds to the formation of the dualDKS comb.
Furthermore, as MgF is birefringent ( n o ∼ . and n e ∼ . at 1554 nm), and the axis of rotation of theWGM resonator is oriented along the optical c-axis, twoorthogonally polarized modes feature a greater differencein their free-spectral range. Note that the material groupvelocity dispersion is anomalous in both direction. Wedemonstrate the generation of two co-propagating solitonstates in orthogonally polarised modes (resonance sepa-ration f m = 1 . GHz). The simultaneous pumping ofboth modes is achieved by aligning the polarisation of thepumps at ◦ with respect to the polarisation of each re-spective mode. Note that half of the energy of each pumpis unused in that case. The pumping efficiency could beimproved by first splitting the laser light with a polarisingbeam splitter, modulating one path with the SSB, andcombining both paths before coupling to the resonator.In this way, each pump can be aligned with the respec-tive polarisation mode, allowing for more efficient cou-pling and avoid energy loss. The resulting combs havea repetition rate difference of ∼ MHz (Fig. S.2c),which is too high for the available bandwidth (very smallcompression factor m = 106 ). As a result, the RF combis heavily aliased at baseband frequency and with the im-age comb centred around f rep − f m , which prevents anyapplication without optical filtering to reduce the band-width. Pumping orthogonal mode can also be realised incounter propagation by selecting the proper pump polar-isation for each direction. Intermodulation products via inter-comb FWM O p t i c a l po w e r d B / d i v . -5 0 5 MHz + 12.4127 GHz d B / d i v . -2 -1.5 -1 -0.5 0 0.5 1 1.5 2GHz + 4.91191 GHz R F po w e r d B / d i v . O p t i c a l po w e r d B / d i v . -50 0 50 MHz + 12.467 GHz d B / d i v . R F po w e r d B / d i v . ab cd Figure S.2. (a)
Generated dual-comb optical spectrum in the co-propagating direction. One of the combs corresponds to atwo-soliton state and hence has a distinct spectral interference pattern. The repetition rate difference is shown in the inset. (b)
Corresponding RF heterodyne comb. (c)
Generated dual-comb optical spectrum in co-propagation, for mode families withorthogonal polarization. The repetition rate difference is shown in the inset. (d)
Corresponding RF heterodyne comb. Themodulation frequency is indicated. The RF comb is heavily aliased due to the insufficient bandwidth for such large repetitionrate difference. for co-propagating solitons — The two generated dis-sipative Kerr solitons, exhibiting different free spectralrange, can in general interact and cause intermodulationproducts. Intermodulation of two co-propagating soli-tons can occur via four-wave mixing, and can lead toadditional sidebands around the optical comb lines. Weconsider here the comb line frequencies in the case of twocombs (1) and (2): ω (1) µ = ω p + µ ω (1)rep (1) ω (2) η = ( ω p + ω m ) + η ( ω (1)rep + ∆ ω rep ) (2)where ( µ, η ) are the azimuthal mode numbers (relativeto the pumped mode, for which µ = 0 and η = 0 ), ω p the pump laser frequency, ω (1)rep the repetition rate of thefirst comb, ω m the single sideband modulation frequencyand ∆ ω rep = 2 π ∆ f rep the difference in repetition rate.Inter-comb four-wave mixing can occur for lines fulfill-ing the phase matching condition (i.e. angular momen-tum conservation that is (cid:82) dφ · e i ( µ + η − µ (cid:48) − η (cid:48) ) · φ ) = 1 ): µ + η = µ (cid:48) + η (cid:48) . For counter-propagating solitons in dis-tinct mode families, this momentum matching cannot besatisfied. However, for two co-propagating mode fami-lies for example, µ + η = ( µ −
1) + ( η + 1) is a possiblepath that conserves momentum, and the resulting fre-quencies are ω (1) µ + ω (2) η = ω (1) µ − + ( ω (2) η +1 − ∆ ω rep ) . As thelast frequency does not coincide with an existing combline, and falls outside the cavity resonance, the mixingproduct is expected to be inefficient (and suppressed bythe cavity lorentzian). Another series of FWM processesleading to the creation of the intermodulation sidebandat ω (1) µ +∆ ω rep are represented in Fig. S.3a, when consid- ering 5 comb lines (the cavity filtering is not accountedfor).The presence of sidebands spaced by ∆ ω rep aroundeach of the comb lines can induce optical-to-RF map-ping ambiguities, as illustrated in Fig. S.4. The beatbetween lines ω (1) µ and ω (2) µ results in the frequency ω (2) µ − ω (1) µ = ω m + µ ∆ ω rep . However, the beat betweenthe adjacent lines ± µ and their sidebands as well as a pairof sidebands around µ ± will be at an identical frequency.Therefore, importantly, the presence of sidebands aroundthe comb lines does not appear in the RF dual-combspectrum, but can be evidenced by the appearance ofseveral lines spaced by ∆ ω rep around the repetition ratesof the combs (or by a high resolution recording of theoptical spectrum). Note that in the present experimentsthe cross products of a comb line and a sideband are at-tenuated by the relative amplitude of the sidebands i.e.at least 20 dB.Experimentally, we evaluated the strength of the in-termodulation sidebands by beating several lines of eachsoliton comb with another reference laser centred at1556.5 nm (Fig. S.3c), having an optical linewidth of ∼ kHz. If the solitons are co-propagating (Fig. S.3d-f), sidebands at ∆ ω rep can be clearly identified. Thismeasurement was repeated while pumping a different se-lection of mode families, such that the scaling of the side-band strength with ∆ ω rep could be retrieved. The resultshown in Fig. S.3d reveal that the mean power of thefirst sidebands (averaged over multiple comb lines) de-creases for larger repetition rate difference, with a slopethat matches a lorentzian profile with a typical linewidthof 170 kHz (full width half maximum) which is in line3 f rep (Hz)-50-40-30-20-10 R e l . s i deband po w e r ( d B c ) dataLorentzian 200 kHz -10 0 10(f-f beat )/ f rep f rep =356 kHz -4 -3 -2 -1 0 1 2 3 4(f-f beat )/ f rep R F PS D ( d B H z - / d i v . ) f rep =738 kHz -2 -1 0 1 2(f-f beat )/ f rep f rep =9.26 MHz Soliton combs CW LaserOBPF
OSA ESA PD p o w e r ( a r b . un i t s ) p o w e r ( a r b . un i t s ) ab de f gc Figure S.3.
Intermodulation of co-propagating solitons (a)
Illustration of FWM processes leading to the formation of the +∆ ω rep sideband around the comb line ω (1) µ (five comb lines are considered here, the cavity filtering is not taken into account). (b) Processes leading to the formation of the +2∆ ω rep sideband around the comb line ω (1) µ (c) Experimental scheme used tomeasure the optical lineshape of the comb lines. Individual lines of each comb are heterodyned with an independent laser, afterselection with an optical bandpass filter (OBPF) and the beat measured on an electronic spectrum analyser (ESA). (d)
Scalingof the mean relative power in the first sidebands (at ± ∆ ω rep ) averaged over several comb lines, for different repetition ratedifference ∆ f rep = ∆ ω rep / π . (e) Heterodyne beatnotes of the reference laser with a line of two co-propagating combs with arepetition rate difference of ∆ f rep = 738 kHz (RBW 50 kHz). The black arrows show the intermodulation sidebands. (f ) Samemeasurement in the case of ∆ f rep = 9 . MHz (RBW 100 kHz). (g)
Heterodyne beatnotes of the reference laser with a line oftwo counter-propagating combs with a repetition rate difference of ∆ f rep = 356 kHz (RBW 50 kHz). optical spectrumoptical frequencies p o w e r ( d B ) p o w e r ( d B ) RF spectrum
Figure S.4. Illustration of the mapping ambiguity inducedby the intermodulation sidebands around the comb lines. Thelines ω (1) µ and ω (2) µ beat at a frequency ω m + µ ∆ ω rep , whichis the same frequency as the beating between adjacent lines/ sidebands. The pairs of optical lines / sidebands beatingat the same frequencies are marked with identical colour (topleft, the spacing of the sidebands was expanded for visual-ization) and their corresponding mixing in the RF domain inindicated by a dot with the matching colour (top right). with the measured quality factors of the resonances.Conversely, when measuring the optical lineshape ofcounter-propagating comb lines, no detectable signs of intermodulation products could be observed in any ofthe ∆ ω rep configurations (Fig. S.3g). In that case, thephase matching condition cannot be fulfilled at the sametime as the energy conservation unless ∆ ω rep = 0 . Stability — In our experiments, the detuning of thelaser with respect to one of the pumped modes is activelystabilised via an offset Pound-Drever-Hall (PDH) lock .This ensures that the resonance-laser detuning remainswithin the soliton supporting range , as the resonatoris free-running and subject to temperature drift. Withregards to the relative stability of the produced dual-comb, this means that the main source of instability isthe drift of the repetition rates difference ∆ f rep , since thefrequency offset between the two pumps is set via electro-optic modulation. We counted the repetition rates oftwo counter-propagating combs and performed an Allandeviation analysis. Up to 10 ms, the repetition rates areaveraging down, meaning that coherent averaging can beperformed up to this duration. On longer timescales,thermal drifts dominate, but we believe that the stabilitycan be easily improved via a thermal stabilisation schemebased on the measurement of ∆ f rep . Triple comb interferogram — We illustrate here an-other triple comb state and show time-domain-based4 -5 -4 -3 -2 -1 Averaging time (s)10 A D EV ( H z ) Figure S.5. Overlapping Allan deviation of the repetitionrates ( ∼ GHz) of two counter-propagating combs. Themarker colours corresponds to each repetition rate. measurement of a triple comb interferogram. We employthe counter-propagating configuration, and set the mod-ulation frequencies to f m = 2 . GHz and f (cid:48) m = 3 . GHzon the modulator (Fig. 3). In that case, the CW combs(1 and 2) correspond to single-soliton states, while theCCW comb 3 results from multiple solitons and fea-tures a complex modulation of the spectral envelope(Fig. S.6a,b). The soliton comb 1 is heavily impactedby a modal crossing on the short wavelength side, whichdecreases its bandwidth. Heterodyning the combs cre-ates a set of three RF combs centred at f m , f (cid:48) m and | f (cid:48) m − f m | = 828 MHz and with a line spacing of 373 kHz,761 kHz, and 1.13 MHz respectively (Fig. S.6d,g,j). Thetime domain interferogram was also acquired. The in-terferogram corresponding to heterodyning each pair ofcombs is retrieved after applying a bandpass filter to se-lect the corresponding RF comb (Fig. S.6e,h,k). The en-velope of each interferogram is also computed. One can note that the strong dispersive wave of comb 1 appearsclearly in the modulated background in Fig. S.6f. Sincetwo of the soliton states (1 and 2) contain a single soli-ton, they can be used to image the number of solitonsand their relative position φ i within a cavity roundtrip inthe comb 3 (Fig. S.6f,i,j). In Fig. S.6i, it appears clearlythat the comb 3 contains 8 solitons. To cross validateour position detection method, we compare the exper-imental optical spectrum of comb 3 with an analyticalexpression for N = 8 solitons with the relative positions φ i / π ∈ [0 , . , . , . , . , . , . , . ,then the identical solitons circulating in the resonatorproduce a spectral interference on the single soliton spec-trum following: S ( N ) ( µ ) = S (1) ( µ ) N + 2 (cid:88) j (cid:54) = l cos (cid:16) µ ( φ j − φ l ) (cid:17) . (3)Here φ i ∈ [0 , π ] is the position of the i-th pulse alongthe cavity roundtrip (assuming a roundtrip normalisedto π ), µ is the comb mode index relative to the pumplaser frequency and S (1) ( µ ) is the spectral envelope of asingle soliton following an approximate secant hyperbolicsquared: S (1) ≈ A sech (cid:18) µ ∆ µ (cid:19) , (4)where A is the power of the comb lines near the pumpand ∆ µ is the spectral width of the comb (in unit of comblines). The expression (3) is computed with the retrievedsoliton positions and the parameter A and ∆ µ are ad-justed to fit the experimental comb amplitude and width.This analytical reconstruction is plotted on Fig. S.6b forcomparison. The complex spectral interference patternis faithfully reproduced, which validates the accuracy ofour dual comb imaging technique. Herr, T. et al.
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Science , 357–360 (2015). O p t i c a l po w e r d B / d i v . CF 12.407 GHz d B / d i v . R F po w e r d B / d i v . -200 -100 0 100MHz + 828.210 MHz R F po w e r d B / d i v . -100 -50 0 50MHz + 2.749 GHz R F po w e r d B / d i v . -100 -50 0 50 100MHz + 3.578 GHz R F po w e r d B / d i v . Time (1 µs /div) V o l t age ( a r b . un i t ) A m p li t ude ( a r b . un i t ) Time (1 µs /div) V o l t age ( a r b . un i t ) A m p li t ude ( a r b . un i t ) Time (1 µs /div) V o l t age ( a r b . un i t ) A m p li t ude ( a r b . un i t ) a bcd e fg h ij k l Comb 2 Comb 3AnalyticalmodelComb 1 + 2Comb 1 + 2 Comb 2 + 1Comb 2 + 3 Comb 2 + 3 Comb 3 + 2Comb 1 + 3 Comb 1 + 3 Comb 3 + 1 Co m b Figure S.6.
Triple comb interferograms (a)
Optical spectrum of the two CW co-propagating combs 1 and 2. The insetshows the three repetition rates of each comb. CF: centre frequency. (b)
Optical spectrum of the CCW comb 3 (green). Themulti-soliton state spectrum is reconstructed after estimating the soliton number and position via dual comb imaging and usingthe analytical expression (3) (grey, the trace is shifted by + 20 dB for visualisation). (c)
Broadband spectrum of the triplecomb interferogram. The spurious peaks in-between the RF combs arise due to the presence of modulation harmonics in theSSB modulated light signal that beat with the comb lines. (d,g,j)
Zoom in of each dual comb RF spectrum (e,h,k)
Timedomain view of each interferogram (after selection with a bandpass filter). (f,i,l)(f,i,l)