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Slow-light analogue with a ladder of RLC circuits J.-P. Cromières and T. Chanelière LaboratoireAiméCotton, CNRS,Univ. Paris-Sud,ENSCachan,UniversitéParis-Saclay, 91405 Orsay Cedex, France [email protected] January 18, 2017 7 Abstract 1 0 2 ThelinearsusceptibilityofanatomicsampleisformallyequivalenttotheresponseofaRLCcircuit. Weusealadderof lumpedRLCcircuitstoobserveananalogueofslow-light,awell-knownphenomenoninatomicphysics. Wefirstchar- n a acterizetheradio-frequencyresponseofthecircuitinthespectraldomainexhibitingatransparencywindowsurrounded J bytwostronglyabsorptivelines. Wethenobserveadelayedpulsewhosegroupdelayiscomparabletothepulseduration 7 correspondingtoslow-lightpropagation. Thelargegroupdelayisobtainedbycascadinginaladderconfigurationdoubly 1 resonantRLCcells. ] h I. Introduction slow-light experimentsperformedinatomicmedia. p - More precisely, Garrido Alzaret al. [2] discussed s s Slow-light is a fascinating phenomenon that has acoupledRLCcircuitsconfiguration toproducean a beenobservedinavarietyofatomicsystems. When EIT-likespectrum. Thisshouldproduceagroupde- l c light passesthrough a medium with a large disper- lay in the time domain. We here propose to cas- . s sion, its group velocity is greatly reduced. Light cade elementary cells to increase the group delay. c has been slowed to 17 m/s [11] and even stopped The idea is quite intuitive but has not been con- i s during a minute [13]. Despite its name, this phe- sidered in details. In the main part of the paper, y h nomenon should notbe restrictedtothe opticaldo- we study a ladder configuration on a factual basis p main. Slow-light is due to a steep dispersion re- and observe a net group delay. Different network [ gion where the group index is heavily modified topologies than a ladder could certainly be consid- 1 as in strongly absorbing media in the vicinity of ered(bridged-Torlatticeforexample). v resonances. Atomic vapors in the optical domain AsaperspectiveintheappendixA,weintroduce 6 are naturally well adapted because the absorbing a general representation derived from a transmis- 5 linesarequitenarrowandopticallythickensembles sion line model (telegrapher’s equations). A de- 6 4 can be obtained. We propose to investigate these scription ofadiscreteseriesbyapropagationequa- 0 twoconditions, narrowresonantlinesandoptically tionhasbeenalreadyapproachedbyNakanishi[18] . 1 dense sample, by using a ladder of lumped RLC with an array of amplifiers for slowly varying DC 0 circuits. signal. The conditions of application have to be 7 A number of different electronic circuits have clarifiedthusdefiningthepropagationconstant,the 1 : beenproposedfordisplayingslow-light. Themajor forwardandbackwardwaves,thecharacteristicand v pioneering experiments were performed by Nakan- inputimpedancesoftheline. i X ishi et al. [17, 15, 18]. Their approach used cas- We propose to produce two closely spaced nar- r caded active low-pass filters and showed a very row resonances. As pointed out by Khurgin in his a clearslow-lighteffect. GarridoAlzaretal. [2]useda remarkable review [14], the so-called double res- doublyresonant(coupledRLC)electroniccircuitto onance scheme can be understood easily and has simulate electromagnetically induced transparency beenusedverysuccessfullytoobverseslow-lightin (EIT), a slow-light precursor that has been exten- atomic medium [6]. The paper is organized as fol- sively studied in multi-level atoms [10]. The work lows. We firstshowthatdoublyresonantRLCcells presented here is inspired by these previous exper- canbe cascadedina ladderconfiguration to mimic iments, but takes a different approach. First, we the increasing absorption and dispersion of an ex- use passive RLC circuits as the elementary blocks tended atomic medium. We then realize the pre- in our slow-light system, in contrast to the active vious scheme with commercial lumped RLC com- filters used by Nakanishi [18]. Passive RLC circuits ponents. The electronic circuit is characterized in havetheadvantagetobeformallyequivalenttotwo- the spectral domain. We finally observe a retarded level atoms in their ground state as many of the pulse whose group delayis comparableto its dura- 1 Slow-lightanaloguewithaladderofRLCcircuits Figure1: Ladder of double resonance cells. Each cell is com- Figure2: Theinputvoltageandcurrentof thecellnumbern posedoftwoseriesRLCcircuitswith(R1,L,Ca)and (R1,L,Cb)placedinparallel. Thedifferentcellsare areVn andIn respectively. Theoutputvoltageand cascadedwithasimpleresistivecouplingR0. current are then Vn+1 and In+1. Z0 and Z1 are genericcompleximpedances. tion. Inthe appendix,wediscussthe applicationof whereω isthesignalfrequency. atransmissionline modelforour configuration. The outputof N cascadedcellscanindeedbede- rivedformallyfromKirchhoff’slawsbywritingthe II. A ladder of lumped RLC circuits transfermatrix II.1. A ladder of double resonance cells 1+ Z0 1 Toobtainthedoubleresonanceschemeasdescribed An =(cid:20) ZV0Inn (cid:21) = " ZZ10Z1 1 #(cid:20) ZV0In+n+11 (cid:21) (2) by Khurgin in [14], we propose to use two RLC cir- = MA (3) n+1 cuits with slightly different resonance frequencies composing anelementarycell(seefig. 1). A has the dimension of a voltage so the transfer n Two series RLC circuits with (R1,L,Ca) and matrix M is dimensionless and is a function of ω (R ,L,C )areplacedinparallel. Theyhavedifferent thatweomitforsimplicity. Inputandoutputofthe 1 b 1 1 circuitarerelatedbytheexpression: resonant frequencies ω = and ω = a √LC b √LC a b respectively but the same linewidth Γ = R1. To A0 = MNAN (4) L carryontheanalogywiththeatomicvaporusedby The exponential MN is calculated by diagonaliz- Camacho et al. in [6], our two RLC circuits would ing M = PDP 1 with − represent the D hyperfine states of caesium. The 2 rheibsoitnaantrcaensssphaoreunldcybweinsudfofiwciebnettlwyeseenptahreamtedleatodienxg- P = ZZ01Λ+ ZZ10Λ− (5) " 1 1 # to slow-light. In other words, each RLC resonator shouldhaveahighqualityfactorsothatΓ ω ,ω . and We can show that by cascading the numb≪er ofa eleb- D = 1+Λ+ 0 (6) 0 1+Λ mentarycells,thetransmissionmimicsthebehavior (cid:20) − (cid:21) ofadoublyresonantatomic medium. where II.2. Lumped element modeling 2 1 Z Z Z Λ = 0 0 +4 0 (7) Theladdercircuitinfig. 1canbemodeledbyusing ± 2Z1 ±s(cid:18)Z1(cid:19) Z1 atransfermatrixforthecellnumberedn(nranging   from1to N where N isthetotalnumberof cells). The boundary conditions are imposed on one The transfermatrix modelin fig. 2is generalbut side by the applied input voltage and on the 0 V in our case we have Z0 = R0 and Z1(ω) is the par- other side by the high measurementload imposing allel impedance (written //) of the two resonators =0. N I R1,L,Ca and R1,L,Cb namely Thevoltagetransferfunction VN isthengivenby 0 V the firstcoefficientof MN = PDNP 1: 1 − Z (ω)= R +jLω+ 1 1 jC ω (cid:18) 1a (cid:19) (1) Λ++Λ //(cid:18)R1+jLω+ jCbω(cid:19) VN =V0Λ+(1+Λ+)N−Λ−−(1+Λ−)N (8) 2 Slow-lightanaloguewithaladderofRLCcircuits Thevoltagetransferfunctiondependsonω. This Withinthetransparencywindowboundedbythe defines the total transmission of the circuit i.e. its two resonances, the slow-light effect can be evalu- Bodediagram. Itiscomplexwithanamplitude(ab- ated by calculating the group delay. This later is sorption) and a phase (dispersion). We will now deducedfrom the dispersion curve (fig. 3.b) by cal- showthatthetopologyproposedinfig. 1produces culatingtheslope(firstderivative)ofthephasespec- two closely spaced absorption peaks. Within the trum. Thecenterofthewindowismarkedbyared transparencywindow, the group delayincreases as square in fig. 3.b. The group delay scales linearly thepeakabsorption. withthenumberofcells N essentiallyfollowingthe increasingpeakabsorption(fig. 3.c). II.3. Absorption spectrum and group de- A ladder of double resonance cells resembles an atomic sample with two neighboring resonances. lay The resonant absorption coefficient goes linearly The expressions (7) and (8) allow to calculate the with the number of cells. Within the transparency transmission spectrum of the ladder in amplitude window,asexpectedfromthedispersioncurve,the and phase. For the numerical application, we use pulsepropagationwillberetardedbythegroupde- the fitted values of R0,R1,L,Ca and Cb that will be lay. Thislatterfollowstheresonantabsorptionasin discussedlaterinIII.1. anatomicvapor. Thedoubleresonancescheme[14] has the advantage of the simplicity. Its main draw- 0 0 back comes from the imperfect absorption at the transparency window center (see fig. 3.a) because B) B) -50 (d -50 of the off-resonance excitation of the neighboring n(d max. transitions. ThisisnotthecasefortheEITinwhich Absorptio-100 NN==16 bsorption-100 ttrheersefoetnroaetnnalcceetriasncnhsaepmaΛreel,neavcyetlriassdcehen-eosmufferes[hd1o0bu]y.lddInebsetthrfueocudtniovdue.bilnAe- -150 N=12 A-150 high resonant absorption is necessary to obtain a a) N=24 c) large group delay in the limit of the absorption of 500 1000 1500 0 10 20 the slow-down pulse because of off-resonance exci- 3π 0 tation. This trade-off guides our experimental approach. 2π ad) π µy(s) -5 III. Experiment Phase(r -0π NN==16 Groupdela-10 III.1. Electronic circuit -2π N=12 We have designed our slow-light circuit using com- b) N=24 -15 d) mercially available components. We decided to -3π 500 1000 1500 0 10 20 implement our slow-light circuit for resonance fre- Frequency(kHz) NumberofcellsN quencies in the MHz range. This choice is based on a number of practical considerations. First, the Figure3: Absorption (a) and phase (b) spectra of the RLC resistance for each element should be kept low to ladderforanincreasingnumberof elementarycells N=1,6,12and24. Peakabsorption(c)correspond- obtain narrow resonances, but it cannot be too low ing to the square purple markers on the absorption because otherwise too much current is drained out spectrum for an increasing number of elements N. of the function generator. A fewohms per element, The group delay in the transparency window (red givingafewtensofohmsforthecircuit,seemsrea- marker on the phase spectrum) isplotted as a func- sonable and compatible with the standard output tionof N. impedanceofagenerator,typically50Ω. Assuming a few ohms for the resistance and a By plotting the absorption spectrum (fig.3.a), we MHz frequency will give an inductance in the µH first observe the two resonances corresponding to range and a capacitance in the nF range. Thus, the resonant excitation of R ,L,C and R ,L,C . a quality factor larger than 10 is expected. From 1 a 1 b The peak resonant absorption increases exponen- a practical standpoint, this range is good to work tially as the number of elementary cells (linear de- in because there is a wide range of commercial in- pendency in log scale of fig. 3.c). Cascading the el- ductors at µH and capacitors at nF. A nF capaci- ements in a ladder configuration mimics an atomic tance is sufficiently high that it will dominate any vapor whose absorption scales exponentially with parasitic capacitances, such as in the BNC cables. the length of the medium (Bouguer-Beer-Lambert Meanwhile, a µH inductance is sufficiently small attenuationlaw). that large DC and RF resistance (due to the wind- 3 Slow-lightanaloguewithaladderofRLCcircuits ingskineffect),whichcanreducethequalityfactor, inputofthecircuitwithaconstantamplitude(typi- are avoided. For our circuit we chose commercial cally100mV).Theoutputwasrecordedwithastan- inductors (Panasonic ELC08D270E)with a uniform dard oscilloscope allowing to measure amplitude inductance of 27µH and just 0.7% RMS dispersity. andphase. Thisvalueismuchlowerthanthecommercialtoler- ance grade. It is important to verify that the inho- 0 mogeneity is low for inductors as they are known tobelessprecisethancapacitors. dB)-20 ( pRF0B=aansed4d.7CoΩn,t=Rh1e1s≃e00p02r.a7pcΩFti.c+Talh4ceΩo,nRsLid=vearla2ut7eiµoHnins,c,lCwuadee=csh6oth8see0 bsorption--6400 b 1 A series resistance of the inductor ( 4Ω), which we ≃ -80 measured independently at 1 MHz. This is mostly 500 1000 1500 2000 due to the skin effect at this frequency. For this set of values, the resonances are expected at ω = b 2π 969kHzand ω =2π 1174kHz. × a × π With respect to the number of elements N, we cmheonstear1y2cpealilrssinoftCotaala.ndNC=b, m12earenpinregseNnt=s a12goeoled- ad) π/2 trade-off as simulated in fig. 3: a 7µs group delay e(r 0 s a anda14dBresidualoff-resonantabsorptionareex- Ph -π/2 pected. The soldering of 24 RLC elements circuit canbedonewithoutspecificexperienceinelectron- -π ics(seefig. 4). 500 1000 1500 2000 Frequency(kHz) Figure5: Absorption (top) and phase (bottom) spectra of the RLC ladder for N = 12 cells. The frequency was swept from 400kHz to 2MHz, covering the region of interest. Theamplitude (top) and thephase (bottom) of theoutgoingsinewave weremeasured (red squares). The blue lines represent the theoretical curves from (7) and (8) using the fitted values of Rfit,Rfit,Lfit,Cfit andCfit (seetext). Thegreenline 0 1 a b (top) is deduced from the propagation equation dis- cussedinappendixA(eq. 26). In the transmission spectrum (fig.5, top), we clearly observe two resonances at ωfit = 2π 1055 a × kHz and ωfit = 2π 1330 kHz. They are both b × deeply absorbing, 82dB. The half-width at half- Figure4: Picture of a ladder of double resonance cells with maximum are typically 50 kHz. We also plot the N = 12. Inductors are prominent as compared to phase of the outgoing sine wave compared to the thelowprofileofthesurface-mountresistorsandca- phase of the incoming one (fig.5, bottom). Close to pacitors. 12inductorsarevisible,theothersaresol- the resonances, the phase is extremely difficult to deredonthebackofthecircuit.Thedistancebetween measurebecauseofthehighattenuation. inductorshas been chosen to avoid direct inductive The measured resonances ωfit = 2π 1055 kHz coupling(topandthebackrowsarestaggered). and ωfit = 2π 1330 kHz devaiates fro×m the ones b × expected (2π 969 kHz and 2π 1174 kHz) from The circuit can now be characterized by measur- × × the nominal values of the components. The devia- ing its transfer function (Bode diagram) and com- tion is most likely due to the parasitic capacitances paredtothemodeldevelopedinII.2. ofthecircuit. Inordertoproperlyfittheexperimen- taldata,weleaveRfit, Lfit,Cfit andCfit asfreefitting 1 a b III.2. Characterization of the spectral re- parameterstoadditionallyaccountfor,forexample, the internal resistance of the inductor and the ad- sponse ditional inductance and capacitance introduced by We use a Tektronix AFG3021B RF function gener- the cables. The value of Rfit was fixed at its nom- 0 ator as a tunable source to record the Bode dia- inal value 4.7Ω (see III.1). The solid blue lines in gram. A continuous sine wave was applied at the fig.5 are the result of this computation. We repro- 4 Slow-lightanaloguewithaladderofRLCcircuits duce fairly well the experimental results with the parameters Rfit = 6.7Ω, Lfit = 25µH, Cfit = 570 1 a pF and Cfit = 910 pF, sufficiently close to the spec- b ification values. From these fitting parameters, we can extract a more accurate value of the linewidth, Rfit Γ= 1 =2π 43kHz,correspondingtoQ-factors Lfit × of 25 and 31 for the resonances ωfit = 2π 1055 b × kHzand ωfit =2π 1330kHzrespectively. a × Except close to the resonances ωfit and ωfit, the b b phase changes smoothly, following the curve pre- 1 1 dicted from eq.(8). In particular, within the transparency window, the phase changes quasi-linearly, u.) 0.5 0.5 a. and the slope directly gives the group delay. A ge( 0 0 a steep variation of the phase (much larger than 2pi Volt−0.5 −0.5 inourcase,seefig.5.b)ispossiblebecauseoftheac- −1 −1 cumulation through many cells. As a comparison, −20 −10 0 10 20 30 −20 −10 0 10 20 30 for a single RLC cell, the phase variation cannot be larger than 2π. This statementis intimately related 1 1 to the slow-light effect because the group delay is u.)0.8 0.8 preciselythefirstderivationofthe phasevariation. a. ( Based on this data, we can expect to see slow- ppe0.6 0.6 lightinthetimedomain. nvelo0.4 0.4 E0.2 0.2 0 0 −20 −10 0 10 20 30 −20 −10 0 10 20 30 III.3. Group delay measurement To observe slow-light, we sent a short pulse into 1 1 the circuit at the center of the transparency win- u.)0.8 0.8 a. dows where the dispersion curve is linear. The y(0.6 0.6 pulse was chosen to be as short as possible but ntensit0.4 0.4 with a bandwidth that fit within the transparency I0.2 0.2 window. Otherwise, pulse distortions would be −020 −10 0 10 20 30 −020 −10 0 10 20 30 Time(µs) Time(µs) expected. Longer pulses can be used as well but if they are too long compared to the measured Figure6: Propagation of a 5µs Gaussian pulse (incoming in delay, the visual impression is less demonstrative. blackandoutgoinginred). Twodifferentcarrierfre- So we chose a 5µs duration, corresponding to a quencies are shown: 840 kHz in the left-hand col- 200kHz bandwidth. This filled approximately 2/3 umn and 1188 kHz in the right-hand column. In of the [ωfit , ωfit] window. The carrier frequency eachcolumn, thetoplinerepresentsthenormalized a b recorded data, the middle line is the pulse envelope of the function generator was set to 2π 1188kHz 1(ωfit+ωfit). The envelope was pr×ogrammed (amplitude envelope of the analytic representation) ≃ 2 a b and the bottom line is the intensity (square of the using the arbitrarywaveform mode of the function envelope). Insteadoftheincomingpulse,theequiva- generator. Theinputandoutputpulsescanbeseen lentfree-spacepropagatingpulsecansometimeserve inblackandredrespectivelyinfig.6. asreference. Inourcase, theyareindistinguishable As a reference, we represent in the left-hand col- because our circuit board is 30cm long, so the free- umn of fig.6 the pulse for a 840 kHz carrier fre- spacedelayistypicallyananosecondwellbelowthe quency(lowdispersion). Weusenormalizedcurves pulseduration. becauseweareonlyinterestedinthedelaymeasure- ment and not the attenuation. It should be kept in mind that the attenuation is far from negligible in the transparency window ( 15dB), so the true am- ∼ plitude at this frequency is much lower than in the referencecase. At the center frequency 1188 kHz (right column of fig.6), the pulse is clearlydelayed,by 7µs for the maximum of the pulse. The pulse is also distorted and elongated from its 5µs incoming duration to 7µs. Distortions are expected because the incom- 5 Slow-lightanaloguewithaladderofRLCcircuits ing bandwidth is not negligible with respect to the going pulse by computing its RMS duration (given transparencywindow. by the fitting procedure) in fig.7 (black curve). We The group delay observed in the experiment is retrieve the 5µs duration of the incoming pulse far substantial,thesamesizeasthepulsewidth,allow- off-resonance (no distortion), while substantial dis- ing the delay to be seen clearly. The delay is even tortions are seen near the two resonances, as well more clearly observed in the intensity data (square asinthetransparencywindow. Aspreviouslymen- of the amplitude envelope), as the intensity pulses tioned, distortionsareexpectedinthetransparency are narrower by a factor of √2 than the amplitude windowbecauseofthefinite incoming bandwidth. pulses. Intensity pulsesarea usefulwayof looking at the data, because this mimics the situation in an III.4. Discussion atomic system, for which intensity measurements are much more straightforward to implement than In our case, the distortion results from the usual fielddetectiontechniques. trade-off observed in optical slow-light and is in- Thesegroupdelaymeasurementsarereproduced dependent of the physical system. As already dis- for different carrier frequencies covering the range cussed,foragivengroupdelay,theincomingpulse 800kHz - 1.6MHz in fig. 7 (red curve). Dispersive should be made asshort aspossible to increasethe delay-to-bandwidthproduct. Thislatterbeingafig- µs)12 ure of merit for slow-light propagation. Distortion ( elay10 appearsbecausethepulse bandwidthisnotstrictly d contained in the transparency window. More pre- p ou 8 cisely,thebandwidthcannotonlyberepresentedby r g alinear dependencyof the phase(givingthe group d 6 n a delay). The first higher order term is due to the ab- tion 4 sorptiononthesideofthetransparencywindow. In a ur 2 other words, the wings of the pulse bandwidth are d S moreabsorbedthanthecenter,asaconsequencethe M 0 R 800 1000 1200 1400 1600 pulseiselongatedinthetimedomain. Ascompared Frequency(kHz) to the double resonance scheme in atomic medium that we keep as a point of comparison [6], the ab- Figure7: Group delay measurements (in red) in the range sorption (or losses) have a different but somehow 800kHzto1600kHz. WealsomeasuretheRMSdu- ration (in black) of the outgoing pulse envelope to comparableorigin. Inourcase,thelossesaredueto characterize its elongation. When the attenuation therestrictivepartofthecircuit,meaningheattrans- is larger than 40dB (shaded areas), the fitting pro- ferintheresistors. Thisisnotthecaseforanatomic cedure is extremely inaccurate, yielding very large vaporinwhichthedecaytermisduetophotonscat- error bars, making the measurements irrelevant in tering. Thisisinterpretedaslossesbecausethescat- this range so they have been discarded. The circles tered photons leave the propagating mode under correspondtothefrequenciesoffig. 6(840kHzand consideration. It should be nevertheless noted that 1188kHz). the atomic decay terms could also represent non- radiative decays as atomic like impurities in solids measurements were extremely difficult when the forexampleleadingtoheattransfer. Sotheanalogy signal was strongly attenuated (> 40 dB), regions betweenatomsandRLCcircuitswithnon-radiative which are represented by the two shaded areas in andresistivelossescouldbepushedonstepfurther. the figure. The fitting procedure givesa poor confi- denceinterval: themeasurementsareirrelevantand have been discarded. The measurements are much IV. Conclusion moreaccurateinthetransparencywindowsandoff- resonance(bothsides). In summary, our setup produces a clearly observ- The red trace in fig. 7 shows that off-resonance, able slow-light effect in a ladder of RLC cells in a for example at 840 kHz (see fig.6, left), the pulse double resonance scheme. The observed delay in exhibits a small delay because the slope dispersion comparable to the pulse duration corresponding to profile is moderate (fig.5, bottom). In the trans- atime-to-bandwidthproductclosetounity. Wealso parency window, meanwhile, the dispersion curve show that cascading elementary cells increases the is steeper yielding a substantial group delay7µs as absorptionanddispersionbothbeingrelatedbythe infig.6(rightcolumn). Thegroup velocitypeaksin Kramers-Kronigrelationduetothecausalcharacter thecenterofthetransparencywindow,demonstrat- ofthe setup. ing clearly that it is a sum of the effects of the two The cascaded elementary blocks provide an in- resonators. teresting resemblance with highly optically absorb- We can finally evaluate the distortion of the out- ing materials. The analogycancertainlybe pushed 6 Slow-lightanaloguewithaladderofRLCcircuits one step further by comparing the structure of the [5] L. Brillouin. Wave propagation and group veloc- Maxwellequations in an atomic [1] and a transmis- ity. Pureandappliedphysics.AcademicPress, sion line model with distributed components [12] 1960. as discussed for a general system of cascaded circuits [21]. We briefly discuss this perspective in [6] Ryan Camacho, Michael Pack, and John How- theappendixA.Thissimilarityinterrogatesthecon- ell. Low-distortion slow light using two ab- ceptof propagation(slow or fast)in structured ma- sorption resonances. 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On the contrary, we’d like to design an ab- A general approach to a finite open transmis- sorbinglinewithresonantlossesasintroducedinII. sionline. EuropeanJournalofPhysics,26(6):1107, We simply generalized the telegrapher’s equation 2005. with arbitrary impedances. A description of our [22] YasuhiroTamayama,ToshihiroNakanishi,and topology in terms of propagation is important be- Masao Kitano. Variable group delay in causeslow-lightisusuallyassociatedtoareduction a metamaterial with field-gradient-induced ∂ω of the group velocity v = in a medium whose g transparency. Phys.Rev.B,85:073102,Feb2012. ∂k response is characterized by a dispersion relation [23] RichardTaubert,MarioHentschel,andHarald (ωistheexcitationfrequencyandkthewavevector). Giessen. Plasmonic analog of electromagneti- Ourgoalinthissectionispreciselytodefineaprop- cally induced absorption: simulations, experi- agationconstantandawavevector. ments, and coupled oscillator analysis. J. Opt. Soc.Am.B,30(12):3123–3134,Dec2013. A.1. Transmission line model [24] Richard Taubert,MarioHentschel, JürgenKäs- Asdiscussed in II.2for the distributed components tel, and Harald Giessen. Classical analog model, the most elementary object is a two-port of electromagnetically induced absorption in component as shown in fig.2. We keep the gen- plasmonics. Nanoletters,12(3):1367–1371,2012. eral case of Z and Z as complex impedances. In- 0 1 [25] Kouki Totsuka, Norihiko Kobayashi, and put/ouput relations for this single component that Makoto Tomita. Slow light in coupled- we wrote in a matrix form (eq.2) arededuced from resonator-induced transparency. Phys. Rev. Kirchhoff’slawsandreadas: Lett.,98:213904,May2007. = Z n+1 n 0 n V −V I [26] YueWang,KunZhang,SongZhou,Yi-HuiWu, 1 = (9) n+1 n n+1 Ming-Bo Chi, and Peng Hao. Coupled-mode I −I Z1 V induced transparency in a bottle whispering- These equations can be extended to continuous gallery-mode resonator. Opt. Lett., 41(8):1825– variablesbywritingV and I asfunctionsof aprop- 1828,Apr2016. agation variable x. The voltage and the current are thenevaluatedatagivenposition x =n dxinthe [27] Herbert G. Winful and Gene D. Cooperman. × Self-pulsingandchaosin distributedfeedback transmission line. Inthisequation, dx isthe charac- bistable opticaldevices. AppliedPhysicsLetters, teristic length of the elementary component, which isintroducedtogivethepropagationvariable x the 40(4):298–300,1982. dimension of a length. The total length of the line [28] Yun-Feng Xiao, Xu-Bo Zou, Wei Jiang, You- is then X = N dx. The correspondence between Ling Chen, and Guang-Can Guo. Analog discreteandcon×tinuous functionsis to multiple electromagnetically induced trans- parencyinall-opticaldrop-filtersystems. Phys. V(ndx)= n V Rev.A,75:063833,Jun2007. I(ndx)= (10) n I 8 Slow-lightanaloguewithaladderofRLCcircuits While we have chosen to introduce dx, the elemen- the two extreme cases: a lossless transmission line tary component length, we could equally use a di- (noresistance)asintheidealtelegrapher’sequation mensionless variablefor x tocontinuously describe the reader may be familiar with, and a purely re- thenumberofelementsn,inwhichcaseV(n) = . sistive line, before moving on to the more complex n V The group velocity would then have the dimension situationrequiredforslow-light demonstration. of inverse time. This is a perfectly valid choice as discussedbyNakanishi[18]. However,wepreferto A.1.1 Lossless transmission line with inductors introduce the unit cell length so that the group ve- andcapacitors locitywillhaveitsstandardunit,lengthdividedby A lossless transmission line is the situation consid- time. ered in the classic telegrapher’s equation. In this The equations (9) for discrete variablescanbe re- case, the impedances Z and Z are purely imagi- placed by their differential counterparts if the dif- 0 1 1 ferential variation from one element to the next is nary, Z = jLω and Z = . If the line is com- 0 1 jCω small. This discretization condition can be written Z posed of discrete components, the continuous solu- 0 1. In what follows, we consider the sit- tion of eq.(13) can be applied if the discretisation Z ≪ (cid:12) 1(cid:12) Z u(cid:12)atio(cid:12)n where the discretization condition does ap- 0 (cid:12) (cid:12) condition 1 is satisfied. In this system, this p(cid:12)ly,(cid:12)andeqs.(9)canberewrittenintheircontinuous (cid:12)Z1(cid:12) ≪ form: condition (cid:12)(cid:12)can(cid:12)(cid:12)be re-written ω ≪ 1/√LC, where the quanti(cid:12)ty o(cid:12)n the right side of this inequality is the cutoff frequency of the circuit [21]. In this case, ∂V(x) Z = 0 I(x) the propagationisdescribedbyapurelyimaginary ∂x dx propagationconstant, ∂I(x) 1 = V(x) (11) ∂x Z dx 1 γ = jω LC/(dx)2 (15) We will focus primarily on voltage measure- q ments, which are described by the second order The term LC/(dx)2 can be replaced by the product equation: λ λ of the distributed inductance λ (henries per L C L unitlength)andthecapacitanceλ (faradsperunit C ∂2V(x) Z = 0 V(x) (12) length). ∂x2 Z1(dx)2 Because γ is imaginary, the solution to eq.(13) is composed of forward +jkx and backward jkx Two boundary conditions are imposed to solve − propagating waves, where the wavevector k = this equations. They are given on one side by the (γ) = ω√λ λ . The amplitudes of these waves input voltage 0 and on the other side by the load ℑ L C V are determined by the boundary conditions, and impedanceattheoutput. Asalreadymentioned,we the backward propagating wave can be removed choose a high load impedance, which means that by choosing what is known as a matched load =0. Thegeneralsolutiontoeq.(12)isthengiven N I impedance. In our case we assume a high load by e γx+eγxe 2γX impedance so there is no matching. This has the V(x) =V0 − 1+e 2γ−X (13) advantageofsimplicityasIN =0butitcanbealso − difficult in practice when Z and Z have complex 0 1 wherewehaveintroducedthepropagationconstant expressions. Aswewillseelater,impedancematch- γ = 1 Z0 whose imaginary part defines the ab- ingisnotnecessarytomimicanabsorbingmedium. dxsZ1 sorption and the real part the dispersion. The two A.1.2 Purelyresistivetransmissionline terms in (13) correspond to the forward and back- Theoppositecasetoalosslesstransmissionlineisa wardsolutions inabroadsense. purelyresistivetransmissionline. Wenowconsider The value of this function at the output of the the impedances Z and Z to be due to resistors transmission line (the transfer function of the line) 0 1 with Z = R and Z = R . The propagation con- is 0 0 1 1 stant γ is purely real, resulting in absorption, with V(X) = V01+2e−e−γ2XγX (14) an absorption coefficient α = ℜ(γ) = d1xsRR01. In Thesolutioniscompletelygeneralatthemoment this situation, strictly speaking there is no propa- because there is no restriction on Z and Z ex- gation because k = 0. The propagation constant 0 1 cept the discretization condition; they are complex γ(ω) is still welldefined and the termpropagation impedances, potentially including a frequency de- should be understood in the generalsense of anin- pendence ω. We now apply the generalsolution to put/outputrelation. 9 Slow-lightanaloguewithaladderofRLCcircuits The discretization condition can be written in givesaphasevelocity termsofthisabsorptioncoefficientasαdx 1: the ≪ ω 1 absorption per elementary cell is low. When this v = = (18) φ k √λ λ conditionholds, theoutputvoltageisgivenby L C which has no frequency dependence. Therefore, V(X) = 2e−αX (16) there is no dispersion, and the group velocity is V01+e 2αX ∂ω − v = =v . g φ ∂k In the strong absorption limit, when the total ab- In Section A.1.2, we considered a resistive trans- sorption αX is large, this can be approximated to mission line, which has γ real. This situation dis- zerothordertoyieldasingle exponentialdecay: plays absorption with an absorption constant α = 1 R V(X) =2V0e−αX (17) dxsR01. Again, there is no frequency dependence in this situation and the transmission line displays Again the output of the transmission line de- no dispersion. However, this strong absorption pends on the boundary conditions, which can be limit is interesting because of the link between the modified by adjusting the load impedance. For a dispersionandabsorptiongivenbythewell-known strongly absorbing medium, impedance matching Kramers-Kronig relations. This implies that if the is less critical because the backward term αx is − strong absorption seen in the resistive line is made small. frequency-dependent, a strong dispersion will be By requiring that the absorption per cell in the obtained, and slow-light propagation could be ob- circuit is small while the total absorption is large, served. we have retrieved in eq.(17) an equivalent of the Inthefollowing section, wefirstconsiderthe dis- Bouguer-Beer-Lambert attenuation law [3]. This is persive properties of a single absorption peak in what we also predict using a discrete components a circuit (single resonant cell). Then, we address model as discussed in II.3 (see the exponential law the more complex situation of two neighbouring infig. 3.c) peaks corresponding to the experimental situation However, while we reproduce an exponential de- presentedinII.1. cay (Bouguer-Beer-Lambert law) using a transmis- sionlinemodel,theanalogywiththeMaxwellprop- A.2.1 Dispersionofasingleresonancecell agation equations in a dielectric medium is incom- plete. In an optical medium, the electric field is To generate an absorption peak rather than a flat described in the slowly varying envelope approxi- absorption as in the resistive transmission line, we mation by a first order differential equation [1] as replacetheresistiveimpedanceZ byahigh-QRLC 1 opposedtosecondorderasineq.(12). Thisleadsto circuit and keep Z as purely resistive. With Z = 0 0 asingleexponentialdecayandnotacompositionof 1 R and Z (ω) = R +jLω+ , the solution in e−αx and eαx. This profound difference in terms of 0 1 1 jCω master equation and boundary conditions should eq.(14)hasthesamegeneralformwith: be kept in mind when the analogy between a modified transmission line and an optically absorbing γ(ω)= 1 Z0 (19) mediumisconsidered. dxsZ1(ω) For a reduction of the group velocity, absorption We can define for this system the resonant fre- only is not sufficient. We now discuss how to pro- 1 R duceastrong dispersivepart. quency ω0 = √LC and the linewidth Γ = L1. In the high-Q limit Γ ω and close to resonance 0 A.2. Introducing a non trivial dispersion ω ω0 < Γ), we ca≪n approximate the solution to | − | firstordertoobtainfortheresonance The two situations presented above can be used α to better understand the conditions required for γ(∆) =α(∆)+ik(∆) = 0∆ (20) slow-light. As stated earlier, slow-light vg vφ 1+jΓ ≪ requires large dispersion. Dispersion is given by where ∆ = ω ω is the detuning from the the frequency dependence of the imaginary partof − 0 the propagation constant γ, which is precisely the resonance frequency and α = 1 R0 the on- 0 wavevector k, while absorption is given by the real dxsR1 partof γ. resonance absorption coefficient. In this equation, In Section A.1.1, we considered a lossless trans- we have retrieved an approximated Lorentzian fre- mission line, for which the propagation constant quency response analogous to that of a two-level was purely imaginary, γ(ω) = jω√λ λ . This atomic system. L C 10