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ORIGINAL ARTICLE

Real-time dynamics and cross-correlation gatingspectroscopy of free-carrier Drude slow-light solitons

Heng Zhou1,2,*, Shu-Wei Huang2,3,*, Xiujian Li2,4, James F McMillan2, Chi Zhang5, Kenneth Kin-Yip Wong5,Mingbin Yu6, Guo-Qiang Lo6, Dim-Lee Kwong6, Kun Qiu1 and Chee Wei Wong2,3

Optical solitons—stable waves balancing delicately between nonlinearities and dispersive effects—have advanced the field of

ultrafast optics and dynamics, with contributions spanning from supercontinuum generation to soliton fission, optical event hor-

izons, Hawking radiation and optical rogue waves, among others. Here, we investigate picojoule soliton dynamics in silicon

slow-light, photonic-bandgap waveguides under the influence of Drude-modeled, free-carrier-induced nonlinear effects. Using

real-time and single-shot amplified dispersive Fourier transform spectroscopy simultaneously with high-fidelity cross-correlation

frequency resolved optical gating at femtojoule sensitivity and femtosecond resolution, we examine the soliton stability limits, the

soliton dynamics including free-carrier quartic slow-light scaling and acceleration, and the Drude electron–hole plasma-induced

perturbations in the Cherenkov radiation and modulation instability. Our real-time single-shot and time-averaged cross-correlation

measurements are matched with our detailed theoretical modeling, examining the reduced group velocity free-carrier kinetics on

solitons at the picojoule scale.

Light: Science & Applications (2017) 6, e17008; doi:10.1038/lsa.2017.8; published online 30 June 2017

Keywords: nonlinear kinetics; optical solitons; photonic bandgap materials

INTRODUCTION

Solitons are stable wave packets that emerge from the autono-mous balance between dispersive and nonlinear phase shifts1,2. Theyhave been examined in diverse media including various opticalfibers3–8, photonic crystal (PhC) waveguides9–11, nonlinear photoniclattices12 and resonant cavities13,14. The formation and propagation ofsolitons precisely reflect the physical characteristics of the operatingregimes, with the key dynamics including the Raman-inducedsoliton self-frequency shift3,4, soliton fission and compression5,9,10,four-wave mixing15, optical Cherenkov radiation16–18, superconti-nuum generation4,6,16, optical rogue waves and fluctuations19,20,astronomical-distance interactions of solitons21, self-similarity22 andevent horizon analogs8, among others23. At the same time, advances insilicon photonic platforms, especially with slow-light PhC structures,provide a broad parameter space for soliton dynamics withtheir tailorable dispersion and enhanced Kerr nonlinearity9–11,24–28.The intrinsic two-photon absorption (TPA) property of single-crystalsilicon when pumped with high peak power near-infrared lasersdrives the high density of free carriers and its corresponding Drudeplasma dynamics. This has been shown to induce solitonacceleration10,29,30, soliton compression31 and soliton fission32 inPhC waveguides.

Here, we advance the dynamic studies of slow-light solitons on-chipwith the inherent Drude electron–hole plasma nonlinearities throughsimultaneous real-time single-shot amplified dispersive Fourier trans-form (ADFT) and cross-correlation frequency resolved optical gating(XFROG) spectroscopies. First, a quartic slow-light scaling (at 7% thevacuum speed of light) of the free-carrier kinetics is described based onthe high-fidelity XFROG characterization of solitary pulses. We provideclosed-form analytical solutions together with the numerical modelingand measurements to describe the self-induced temporal accelerationand self-frequency blue-shift. Second, we extend the description of theDrude electron–hole plasma perturbations to femtosecond pulsespectrum broadening and optical Cherenkov radiation. We describethe soliton-to-soliton stability limits of the free-carrier plasma-inducedspectral broadening via single-shot ADFT characterization in real time.In the presence of free-carrier dynamics, the soliton spectrum histo-grams demonstrate near-Gaussian limited intensity distributions withmarkedly high stability. Third, we investigate the free-carrier-perturbedmodulation instability (MI) that occurs in both normal and anomalousdispersions, which has a broader gain spectrum and is thresholdlesscompared to the conventional Kerr MI. Our studies are matched withthe resonance condition of Cherenkov radiation and the modified MIgain spectrum, including numeric and analytical solutions.

1Key Lab of Optical Fiber Sensing and Communication Networks, University of Electronic Science and Technology of China, Chengdu 611731, China; 2Optical NanostructuresLaboratory, Columbia University, New York, NY 10027, USA; 3Mesoscopic Optics and Quantum Electronics Laboratory, University of California, Los Angeles, CA 90095, USA;4College of Science, National University of Defense Technology, Changsha, Hunan 410073, China; 5Department of Electrical and Electronic Engineering, The University of HongKong, Hong Kong, China and 6Institute of Microelectronics, Agency for Science, Technology and Research, Science Park II, Singapore 117685, Singapore

Correspondence: H Zhou, Email: [email protected]; S-W Huang, Email: [email protected]; CW Wong, Email: [email protected]*These authors contributed equally to this work.

Received 3 January 2017; revised 8 February 2017; accepted 8 February 2017; accepted article preview online 10 February 2017

Light: Science & Applications (2017) 6, e17008; doi:10.1038/lsa.2017.8Official journal of the CIOMP 2047-7538/17www.nature.com/lsa

MATERIALS AND METHODS

Figure 1a shows the hexagonal-lattice PhC fabricated with optimizedphotolithography in a foundry. The device is 250 nm thick, and thebottom SiO2 cladding is 3 μm. The photonic bandgap nanomembrane,from an initial silicon-on-insulator wafer, has a periodicity, a, of431 nm and a hole radius of 125 nm (0.290a). A single rowtranslational defect is introduced, creating a 1.5 mm W1 slow-lightPhC waveguide. The slow-light silicon PhC waveguide is then air-bridged by removing the bottom SiO2 with buffered hydrogen fluoride(HF) etch. The group index ng and group velocity dispersion (GVD)of the sample are measured with an optically clocked coherent sweptwavelength interferometry25. In the swept wavelength interferometry,a fiber Mach-Zehnder interferometer with a 40 m path lengthimbalance is utilized to provide the base clock for the equidistantoptical frequency sampling. In addition, the transmission spectrum ofthe hydrogen cyanide gas cell is acquired simultaneously as an absolutewavelength calibration to remove the synchronization errors andensure the proper alignment of successive data sets. The resultingGVD and third-order dispersion are then extracted from the delayspectrogram and are shown in Figure 1a.The pump pulse we use to interrogate the slow-light soliton

dynamics is a 39 MHz repetition rate picosecond laser with a 2.3 psfull-width half-maximum pulse width (nearly transform limited), andits center wavelength is adjustable between 1545 and 1565 nm. A high-fidelity second-harmonic-generation based XFROG system (Figure 1b)is constructed to simultaneously capture the output pulse intensity,phase, and spectrum33,34. Due to the presence of a pre-defined andstrong reference pulse, the temporal direction degeneracy is avoided in

the XFROG measurement and can be used to characterize the weakpulses with sub-fJ sensitivity. In our setup, half of the pump pulse istapped out before entering the PhC waveguide and is utilized as thereference pulse in the XFROG system. The temporal structure of thepump pulse is independently characterized a priori using a standardsecond-harmonic-generation FROG measurement system. The splitand residual pump pulses are coupled into the PhC structure. Thepulse output carries information of the slow-light soliton dynamicsand is treated as the test pulse of the XFROG system. As shown inFigure 1b, our measurements avoid any pulse amplification, pulsedistortion and optoelectronic conversion to ensure high-fidelity solitoncharacterization. Figure 1c shows an example XFROG spectrogram ofthe output pulse and the retrieved temporal pulse shape. To ensureproper convergence, the phase retrieval routine is concluded after theFROG error drops below 5× 10− 3 for all cases reported here34.Furthermore, we set up a single-shot spectrum measurement system

based on the ADFT to study the shot-to-shot fluctuations of thesubpicosecond slow-light soliton spectra at a repetition rate of 39MHz(Refs. 35,36). In the ADFT set-up, a long spool of dispersion-compensating fiber encompassing a normal dispersion of1.33 ns nm− 1 is used to provide the necessary time stretching, asshown in Figure 1d, followed by a pre-amplifier along with a fast near-infrared photodiode–oscilloscope subsystem. The ADFT has anequivalent spectral resolution of 190 pm, and its fidelity is confirmedby comparing the averaged measurements with the spectra obtainedfrom the optical spectrum analyzer (OSA). Before recording the timeseries of up to 4000 spectra, an averaged time-stretched spectrum(100 times) is captured and used as the reference spectrum for the

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Figure 1 XFROG and real-time single-shot spectroscopy utilized to elucidate the Drude free-carrier soliton dynamics. (a) Group indices (data points with fittedgreen solid line), group velocity dispersion (red) and third-order dispersion (blue) of the slow-light PhC, measured via optically clocked swept wavelengthinterferometry. The waveguide transmission is shown in the upper panel (blue curve). The inset is the SEM image of the PhC, and the scale bar=400 nm.(b) XFROG approach using second harmonics. (c) An illustrative XFROG spectrogram of the 1555 nm soliton at the output of the slow-light PhC waveguide.The retrieved intensity waveform is shown in white. The inset is the reference FROG spectrogram for the soliton measurements. (d) The ADFT set-up for real-time soliton characterization. BBO, barium borate nonlinear crystal; BS, beam splitter; DCF, dispersion compensation fiber; FL, focusing lens; PD,photodiode.

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Light: Science & Applications doi:10.1038/lsa.2017.8

cross-correlation analysis. The cross-correlation function is calculatedusing the definition:

Ccorr tð Þ ¼Rr tð Þs t � tð ÞdtR

r tð Þj j2dt R s tð Þj j2dt ð1Þ

where r(t) denotes the reference spectrum and s(t) denotes the single-shot spectrum.To help gain physical insights into the slow-light soliton dynamics,

we adopt the symmetric Fourier split step method to compute thenon-integrable nonlinear Schrödinger equation (NLSE) with anauxiliary equation describing the free-carrier generation26,31:

∂E∂z

¼ �aE þ iXkZ2

bkk!

i∂∂t

� �k

þ i s2geff Ej j2 � sk0kcNc

� �E ð2Þ

a ¼ alinear2

þ s2bTPA2A0

Ej j2 þ sNcs2

ð3Þ

Nc ¼Z T

�N

s2bTPA2hn0A2

0

Ej j4 � Nc

tf

� �dt ð4Þ

Here, s= neff/no is the slow-light factor, βk is the k-th order dispersion,αlinear is the PhC waveguide linear loss, γeff= 2πn2/λA0 is the Kerrnonlinear coefficient, βTPA is the TPA coefficient, Nc is the TPA-generated free-carrier density, τf is the free-carrier lifetime, σ is thefree-carrier absorption coefficient (FCA) and kc is the free-carrierdispersion coefficient (FCD). The parameters used are illustrated inTable 1 and are from either our measurements (linear losses anddispersion of the PhC waveguide) or the published literature26,31,37.The precise input pulse shape is directly retrieved from the FROGmeasurement, providing the exact initial conditions for the NLSEmodeling.

RESULTS AND DISCUSSION

Figure 1a plots the measured wavelength-dependent transmission andgroup index (upper panel), as well as the GVD and third-orderdispersion of the PhC waveguide (lower panel). Within the PhCbandgap between 1540 and 1560 nm, the group index increases from7.5 to 14.5 while the anomalous GVD can still support picosecondsolitons. In this paper, we highlight the most heuristic soliton evolutionresults pumping at 1555 nm (ng= 10.0, β2=− 1.6 ps2 mm− 1,β3= 0.22 ps3 mm− 1) and 1560 nm (ng= 14.5, β2=− 4.5 ps2 mm− 1,β3= 1.0 ps3 mm− 1).Figure 2 shows the retrieved pulse profiles and spectra as a function

of the incident pulse energies. At 1555 nm, where the slow-light factor

is relatively small, the output pulse exhibits flat phase profile andtemporal compression to 2.1 ps. On the other hand, with increasingslow-light factor at 1560 nm, the soliton compression is absent but theoutput pulse exhibits a strikingly large temporal acceleration and self-frequency blue-shift. The temporal acceleration and the self-frequencyblue-shift can be tuned continuously by changing the input pulseenergy (Figure 3a). This leads to a maximum measured temporalacceleration as large as 6.4 ps, which is almost three times the inputpulse width.These measurements deterministically capture the pulse dynamics

with combined temporal and spectral information and thus offer idealplatforms to understand the increased interactions in the large groupindex regime. Consequently, we model the pulse dynamics using thefull-scale NLSE, which includes the slow-light scaling, exact devicedispersion, TPA and dynamics of the free carriers. Slow-lightenhancements in the Kerr nonlinearity and TPA, both of which arerooted in third-order nonlinearity, scale with s2 (where s denotes theslow-light factor) because of the prolonged light–matter interactiontime and increased light intensity because of spatial compression27,37.The slow-light enhancements of the FCD and FCA are, however, moresubtle. First, the generated electrons and holes are from the TPA, andthe carrier density scales quadratically with the laser intensity, whichinherently scales with s2 because of the intensity enhancement37.Second, the accumulation time (T in Equation (4)) and the polariz-ability (s in the last term of Equations (2) and (3)) of the free carriersare both affected by the prolonged interaction time, leading to anothers2 enhancement. This can give rise to an overall s4 scaling of the FCAand FCD. These mechanisms can be viewed under the interpretationthat the time-accumulated free-carrier generation, and the cascadedFCA and FCD provide more intense enhancements due to the slow-light compared with the Kerr nonlinearity, which is based on theinstantaneous electric polarizability38.As shown in Figure 2a–2d, when s4 scaling of the FCD and FCA is

included in the model, remarkable agreements are achieved betweentheory and experiment for the soliton intensity, shape, phase,spectrum and acceleration. Note that the values of s for thewavelengths examined here (s= 2.9 for 1555 nm and s= 4.2 for1560 nm) already reflect an ~70 to ~ 310 times scaling of the Drudefree-carrier plasma effects, which are sufficient to manifest theunderlying physics. To understand the slow-light scaling, we showthe root-mean-square deviations between the experimentally mea-sured and NLSE-calculated pulse intensity profiles and spectra underdifferent scaling rules in Figure 2e and 2f. Unambiguously, with the s4

enhancement for the free-carrier effects and the s2 enhancement forboth the Kerr and TPA, the smallest deviation of the temporalwaveform and the spectra is obtained for a diverse array of solitonenergies. This is especially true at the highest pulse energies thatencounter the largest nonlinear contributions, supporting the quartics4 enhancement scaling of the Drude plasma nonlinearities in slow-light PhCs.Furthermore, numerical modeling clearly shows that both the

temporal acceleration and the self-frequency blue-shift originate fromthe slow-light-enhanced Drude free-carrier effect. Compared with theinstantaneous Kerr effect, the nonlinear phase shift from the FCDresults from the accumulation of free carriers generated from the TPA,resulting in a quasi-linear profile, as illustrated in Figure 3b. Conse-quently, the quasi-linear phase shift leads to a center frequency shift,and the temporal acceleration follows because of the anomalousdispersion. In addition, the cascaded nonlinear nature of the TPA–FCD process lends itself to more slow-light enhancements, leading to amore prominent effect at 1560 nm when the slow-light factor is

Table 1 Summary of parameters used in the NLSE simulations

Parameters Values

Linear loss αlinear 2.5 dB mm−1

Kerr nonlinear index n2 5×10−18 m2 W−1

Effective nonlinear coefficient γeff2pn2A0l

Two-photon absorption βTPA 8.8×10−12 m W−1

Free-carrier absorption σ 1.45×10−21 m2

Free-carrier dispersion kc 1.35×10−27 m3

Modal area A0 0.13 μm2

Free-carrier lifetime τf 500 ps

Soliton number Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2P0geffT

20

b2j j

r

Abbreviations: NSLE, nonlinear Schrödinger equation; TPA, two-photon absorption.

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higher. Moreover, Figure 3c illustrates that increasing the 1555 nmpulse energy to 25 pJ improves the pulse acceleration by up to 6 ps(more than 2.5 × the pulse width) while maintaining a favorable pulseshape with modest compression. Further increases in the pulse energysplit the soliton pulse.To further understand the Drude soliton dynamics under slow-light

enhancement, we next conduct the soliton perturbative analysis underthe TPA–FCD. The detailed derivation of the perturbation is given in

Supplementary Information Section I, and the results can besummarized in the following two closed-form equations for the self-frequency blue-shift and temporal acceleration:

Do ¼ 4

15

k0kcbTPAhn0A2

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z ð5Þ

Dt ¼ � 2

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k0kcbTPA b2j jhn0A2

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z2 ð6Þ

1.0a b

c d

e f

7.5 pJN=2.0

5.0 pJN=1.63

1.65 pJN=0.94

6.8 pJ

Input

7.5 pJN=2.0

5.0 pJN=1.63

1.65 pJN=0.94

Input

N=1.6

5.0 pJN=1.4

1.5 pJN=0.76

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6.8 pJN=1.6

5.0 pJN=1.4

1.5 pJN=0.76

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Figure 2 Quartic slow-light scaling of the Drude free-carrier nonlinear dynamics upon soliton propagation. The solitary dynamics are mapped with differentinput pulse energies and soliton numbers N for two example wavelengths of 1555 nm (ng=10.0) and 1560 nm (ng=14.5). (a,b) For 1555 nm, the blue-shaded area and blue dashed lines are the experimental cross-correlation-retrieved intensity and phase, respectively. The red dashed and solid line plots arethe captured phase and intensity from the numerical predictions, respectively. (c,d) Same as (a,b) but for 1560 nm. In both center-wavelength cases, thesoliton acceleration and self-frequency blue-shifts are observed at various magnitudes. (e,f) Measurement comparison with the pulse evolution modeling viaNLSE including the Drude free-carrier dynamics elucidates the exponent in the slow-light scaling. Panel e is for the 1555 nm soliton and f is for the1560 nm soliton. The top panel is the temporal measurement-theory RMSD. The bottom panel is the spectral measurement-theory RMSD. Letters in thelegend are as follows: F, FCD and FCA dynamics; K, Kerr nonlinearities; RMSD, root-mean-square deviation; T, two-photon absorption.

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Light: Science & Applications doi:10.1038/lsa.2017.8

When the system parameters (Table 1) are entered, the estimatedtemporal acceleration from Equation (6) is 7.6 ps, which matches wellwith the experimentally measured acceleration of 6.4 ps for the 1560 nmpulse at 6.8 pJ. Of note, the effects of the FCD shown in Equation (5)are opposite to the self-frequency red-shift induced by Raman scatteringin nonlinear optical fiber platforms23. While the Raman effect is moreprominent when the system is interrogated by ultrashort laser pulses,the TPA–FCD effect is not changed with the input pulse duration if thefundamental soliton condition is satisfied. Hence, relatively long pulses,such as the 2.3 ps pulse used in this work, are preferred to observe theTPA–FCD effect in the absence of the Raman effect.The Drude FCD also results in perturbations of the spectral

broadening and optical Cherenkov radiation, as depicted inFigure 4. Figure 4a shows the experimentally measured spectraldynamics of the 780 fs (full-width at half-maximum) pulse. Tofacilitate the nonlinear spectral broadening of the femtosecond pulses,we utilize a PhC waveguide with a specifically designed low GVDregime near 1545 nm via symmetric lattice shift39,40 (the GVD curve ofthis waveguide is differentiated slightly from the typical W1 PhCwaveguide shown in Figure 1a, with the details given in SupplementaryInformation Section III). Here, the center wavelength is chosen at1560 nm with ng ~ 14.0 to harvest more slow-light enhancements inthe FCD. As the input pulse energy is increased from 3.5 to 7.0 pJ, the10-dB bandwidths are increased from 12.5 to 15.4, 22.2 and 25.1 nm.The measurements (blue solid lines) are well-matched to the NLSEnumerical model (red solid lines). To identify the origin of theobserved spectral broadening, we artificially turn off the FCD in theNLSE numerical model, and the magnitude of spectral broadening isgreatly reduced (green dashed lines). While the Kerr nonlinearity isquenched by the nonlinear absorption in the silicon waveguide, theDrude TPA–FCD takes over and dominates the spectral broadening inour slow-light waveguide parameters.To investigate the Drude free-carrier plasma-perturbed dynamics of

Cherenkov radiation, Figure 4b shows the NLSE-simulated spectralbroadening kinetics in our waveguide with a 7 pJ pulse energy and anapproximate 780 fs pulse width. The FCD contribution is numericallytuned to evaluate its cumulative impact on the output spectra.Particularly, without the FCD, an isolated Cherenkov radiation side-band is obtained at ~ 1534.0 nm (≈50 dB smaller than the solitonspectrum), based on the zero-crossing of the dispersion near 1545 nm.

When the FCD effect is numerically tuned to be larger, a clear blue-shift in the spectra is observed. Correspondingly, the dispersive wavemanifests a red-shift to counterbalance this effect, cumulating inoverlapped soliton and dispersive wave spectra. The mechanisms forsuch dynamics are multifold: first, a blue-shift of the soliton spectrumvia the FCD corresponds to a red-shift of the phase-matched radiationspectrum18. Second, the blue-shifted soliton spectra overlap betterwith the red-shifted radiation components, leading to a rapid growthin the radiation intensity16. Third, the FCD modifies the phase-matching between the soliton and dispersive waves, enabling con-tributions to both the wavelength and intensity of the dispersive waves.In particular, the phase-matching condition of the Cherenkovradiation6,18 is now modified by the FCD to:

XnZ2

bn oSð Þ oD � oSð Þnn!

¼ |SPM þ |FCD ð7Þ

where βn is the nth order dispersion, ωS and ωD are the angularfrequency of the soliton and dispersive wave emission, respectively,and ϕSPM and ϕFCD are the phase shift due to SPM and the FCD,respectively. Figure 4c shows the calculated phase-matching wave-lengths with different values for the total nonlinear phase shift (ϕSPM

+ϕFCD). First, with zero FCD and a soliton wavelength at 1560.0 nm(green spot 1 in Figure 4c), the analytically estimated phase-matchingwavelength is ~ 1534.0 nm (yellow spot 1 in Figure 4c), which agreeswell with the NLSE simulations (green and yellow spots 1 in Figure 4b,see Supplementary Information Section II). Increasing the FCD causesthe Drude soliton spectrum to undergo a blue-shift (that is, ωS

becomes larger) while the phase shift from the FCD increases (thatis, ϕSPM+ϕFCD equals approximately 2s2γeffPo and 3s2γeffPo for spots 2and 3, respectively), and the estimated resonance conditions aremarked in Figure 4c (spots 2 and 3). These are also consistent withthe numerical simulations shown in Figure 4b (spots 2 and 3,correspondingly) and our qualitative analysis above. Subsequently,with the FCD fully applied, the soliton spectrum and Cherenkovradiation emerge together. Therefore, in spite of the Cherenkovradiation not being distinctly separated from the soliton spectra, ouranalysis notes the underlying contribution and dynamics of theCherenkov radiation perturbed by the Drude FCD. Furthermore,the FCD-induced Cherenkov radiation beneath the soliton spectrum isverified with the modeled time-domain waveform illustrated in

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Figure 3 Influence of the Drude free-carrier dynamics on the slow-light soliton dynamics. (a) Under the Drude free-carrier perturbation, the measured solitonaccelerations and self-frequency blue-shift are shown as the solid line with triangles and dashed lines with circles, respectively. The 1560 and 1555 nmslow-light examples are illustrated in blue and red. (b) The modeled contributing phase shifts are from the Drude free-carrier dynamics, self-phasemodulation and group velocity dispersion. The input is a 1555 nm pulse at 7.5 pJ (N=2.0), with the pulse shape illustrated in the dashed background. Theinset shows the corresponding measured XFROG spectrogram. (c) For a 1.5 mm slow-light photonic crystal waveguide, the maximum acceleration with anacceptable pulse shape is ~9.0 ps. The parameters used in the simulation are identical to those in Figure 2a and 2b. The insets show the correspondingspectral evolutions.

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Figure 4d. Particularly, with the FCD applied, a larger but more slowlyoscillating dispersive wave tail is obtained because of the overlappingof the Cherenkov and soliton spectra, in contrast to the smaller andmore rapidly dispersive wave tail incurred by the far separatedCherenkov and soliton spectra without the Drude FCD.With the spectral broadening induced by the slow-light-enhanced

FCD, the ADFT is implemented to investigate the soliton-to-solitonstability of the broadened Drude soliton spectral components.Figure 5a shows the averaged optical spectra of the spectrallybroadened pulses, measured with a grating-based OSA. The single-shot ADFT measurements are conducted on the pump not only at

1560 nm but also at 1540 nm, especially the spectral corners whereinany unstable features would be expected41. Figure 5b shows therecorded single-shot spectrum waveforms. The ADFT not only fullyretrieves the detailed spectral characteristics but also unveils the real-time spectral dynamics that cannot be acquired in conventional OSAsor RF spectrum analyzers. Furthermore, Figure 5c summarizes thecross-correlation of the time-averaged (reference) spectra from theOSA and the single-shot spectra, where a near Gaussian spread around+1 containing minuscule instability is observed. The computed cross-correlation average values are Mfcd= 0.9989 and Mpump= 0.9978 forthe FCD sideband and near the pump, respectively. Both values ofvariation (Vfcd= 3.4× 10− 8 for the FCD sideband andVpump= 2.0× 10− 7 near the pump) are small and within the allowedinstrumentation noise floor. This demonstrates the stable soliton-to-soliton behavior of the Drude TPA–FCD-induced spectral broadening,capturing the high stability of the Drude free-carrier nonlinearabsorption (σ) and dispersion kc parameters.To further examine soliton dynamics under the Drude free-carrier

plasma, we next theoretically and numerically examine the effects ofthe FCD on the MI gain spectrum. We assume:

E ¼ffiffiffiP

pþ a z; tð Þ

h iei geffP�

k0kcbTPAtf2hn0A

20

P2

h iz ð8Þ

Nc ¼ bTPAtf2hn0A2

0

P2 þ b z; tð Þ ð9Þ

where a and b are the perturbations. Here, the implicit assumption isthat the absorption is not yet so strong that we can assume the steady-state solution has a constant amplitude along the propagation. This isa necessary approximation because no analytic steady-state solutionexists if we consider all the dynamical absorptions. Using the linearstability analysis, we obtain the Drude-perturbed MI gain spectrum as(detailed in Supplementary Information Section IV):

GMI ¼ Im b2Oj jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiO2 þ 4geffP

b2� 4k0kcbTPAP

2

hn0A20b2

1tf� iO

� �vuut

0B@

1CA ð10Þ

It is seen from Equation (10) that in contrast to the conventionalKerr MI utilized in optical parametric oscillators23,42,43, the MI gain isalways guaranteed by the Drude FCD regardless of the MI frequencyand is present in both the normal and anomalous GVDs. Figure 6ashows the numerically calculated MI spectra with a 3 W continuous-wave pump. Agreeing with our analysis, the MI induced by the DrudeFCD features a thresholdless amplification and a broader gainbandwidth. Furthermore, as shown in Figure 6b, in the normaldispersion region, the FCD-perturbed MI is possible, a result that isfundamentally different from that for the canonical Kerr MI, whichonly exists in the region of the anomalous GVD. In both regions, thespectra drop off with the increase in the frequency detuning due to the1/Ω term in Equation (10)44. Moreover, it is seen from Equation (10)that the MI perturbed by the FCD is affected by the free-carrierlifetime (the last term under the square root of Equation (10)), whichis confirmed by the numerical simulations shown in Figure 6c. Inparticular, with a decreased τf, the numerically calculated MI spectrabecome narrower, which is consistent with the trend predicted byEquation (10). With zero free-carrier lifetime, the contribution fromthe FCD completely vanishes and the numerical MI spectrum turnsinto the pure Kerr case. Finally, Figure 6d shows the influence of theTPA on the MI gain spectrum. In particular, for the MI induced bythe Kerr effect, the TPA loss severely suppresses the MI sidebands

Nor

m. s

pect

rum

(10

dB

per

div

ison

)

Nor

m. i

nten

sity

(0.

25 p

er d

ivis

on)

Rad

iatio

n w

avel

engt

h (n

m)

Nor

m. s

pect

rum

(10

dB

per

div

ison

)

1520

1545

1540

1535

15301550 1555 1560

Soliton wavelength (nm) Time (ps)1565 –15 0 15 30

7 pJ

a b

c d

N=3.73

5.2 pJN=3.22

3.5 pJN=2.64

Input

0

0

GV

D (

ps2

mm

–1)

20

–20

3

3 2 1

2

1

1

1

2

3

–2 (14 pJ)

2 (14 pJ)With FCDw/o FCD

3 (21 pJ)

0

23

25%50%75%100%

1540 1560 1520 1540 1560Wavelength (nm) Wavelength (nm)

Figure 4 Drude FCD-based broadened spectra with optical Cherenkovradiation in the femtosecond soliton regime. (a) Experimental (blue) andNLSE simulated (red) spectral broadening with different pump pulseenergies and solitary wave numbers N. Excellent agreement is achievedwhen the quartic slow-light scaling is included in the analysis. Forillustrative comparison, dashed green lines show the NLSE simulated spectrawhen the FCD effect is numerically turned off. (b) The NLSE simulatedspectra and dynamics from the Cherenkov radiation with increasing DrudeFCD coefficient contributions. The estimated Cherenkov radiation peaks atthe yellow spots 1–3 are 1534.0, 1538.5 and 1541.0 nm, and thecorresponding soliton center wavelengths at the green spots 1–3 are1560.0, 1556.7 and 1554.8 nm. The dispersion curve for the simulation isshown on the right vertical axis, with β2=−3.5 ps2 mm−1,β3=0.5 ps3 mm−1, β4=−0.005 ps4 mm−1 and β5=−0.0025 ps5 mm−1,centered at 1560 nm. The zero crossing of the dispersion comes from thesymmetrical lattice shift to build the low GVD region within the passband ofthe photonic crystal waveguide. (c) Analytically estimated phase-matchingcondition for Cherenkov radiation under different effective nonlinear phaseshifts (corresponding pulse energy in the brackets), which are normalized tos2γeff Po. (d) The temporal waveform corresponding to the spectra shown inb, with the FCD both present and absent. The inset is the close-up signal ofthe temporal dispersive wave tails without the FCD, illustrating the fasterand weaker temporal oscillations that arise from the heterodyne beating ofthe dispersive wave far separated from the soliton wave.

Free-carrier Drude slow-light solitonH Zhou et al

6

Light: Science & Applications doi:10.1038/lsa.2017.8

because the Kerr MI originates from the parametric amplification thatis directly proportional to the light intensity. However, if TPA isswitched off for FCD-induced MI, the free-carrier densities drop tozero, collapsing the MI gain spectrum back to its original form. Theuniqueness of the FCD-induced MI dynamics holds potential torealize on-chip broadband laser sources.

CONCLUSIONS

Here, we examined ultrafast solitary wave kinetics in slow-lightsilicon-based photonic bandgap waveguides based on ADFT real-time spectrum analysis and phase-resolved XFROG. With the slow-light-enhanced Kerr nonlinearity and Drude free-carrier effects, wedemonstrated a quartic slow-light scaling of the Drude free-carrier

1.0a b c

0.8

0.6

Nor

m. s

pect

rum

Nor

m. i

nten

sity

Nor

m. i

nten

sity

Cro

ss-c

orre

latio

n

0.4

0.2

0.0191 192 193 194 195 10

0.0

0.5

1.00.0

0.5

1.0 1.000 1.000

0.998

0.996 0.996

1.000

0.996

1.000

0.998

0.996

20 30 40 50 60 70 0 1000 2000 3000 2 0

1 0

Shot number Histogram(counts/100)

Time (ns)Optical freq. (THZ)

Figure 5 ADFT-based single-shot and real-time spectral measurements. (a) Averaged spectrum of the Drude FCD-induced spectral broadening and opticalCherenkov radiation, measured with an OSA. The soliton pulse energy is 7.0 pJ, and the center wavelength is at 1560 nm. (b) Example single-shot spectracorresponding to the red (near the pump wavelength) and green (the edge of the broadened spectrum) highlighted spectral components. The traces arecaptured using a real-time oscilloscope, with the temporal horizontal axis swapped to the optical spectra through the ADFT. (c) The single-shot spectra andtime-averaged spectrum cross-correlation. Histograms of the cross-correlations demonstrate the nearly Gaussian spread around +1 with minuscule variations.The average cross-correlation for the Drude-broadened spectral sideband Mfcd is 0.9989, with a variation Vfcd of 3.4×10−8. The cross-correlation mean valuenear the pump Mpump is 0.9978, with a variation Vpump of 2.0×10−7. Both variation values are small and within the measurement noise floor, demonstratingstable soliton-to-soliton behavior.

Nor

m. s

pect

rum

(20

dB p

er d

ivis

on)

Nor

m. s

pect

rum

(20

dB p

er d

ivis

on)

Nor

m. s

pect

rum

(20

dB p

er d

ivis

on)

Nor

m. s

pect

rum

(20

dB p

er d

ivis

on)

1540

a b

c d

1550

0 1%

1 ps 10 ps100 ps 500 ps0

10% 50% 100% 0 1% 10% 50% 100%

1560 1570 1580 1540 1550 1560 1570 1580

Wavelength (nm)

1540 1550 1560 1570 1580 1540 1550 1560 1570 1580

Wavelength (nm) Wavelength (nm)

Wavelength (nm)

Figure 6 Theoretical study of the concurrent Drude FCD-perturbed MI in silicon photonic crystal waveguides. (a,b) NLSE simulated FCD perturbed MI spectraof a 3 W continuous wave pump for increasing FCD strength, (a) for the anomalous dispersion regime (β2=−2.2 ps2 mm−1) and (b) for the normal dispersionregime (β2=2.2 ps2 mm−1). (c) NLSE simulated FCD perturbed MI gain spectra for different free-carrier lifetimes in the anomalous dispersion regime(β2=−2.2 ps2 mm−1). (d) Comparison of the influence of the TPA loss on the MI gain spectrum with (red) and without (black) the Drude FCD for a 3 Wcontinuous wave pump in the anomalous dispersion regime (β2=−2.2 ps2 mm−1). Solid lines show the cases with the TPA loss, and the dotted line showsthe case without the TPA loss. Note that the solid red line does not have a corresponding dotted red line considering that the FCD must occur together withTPA. FCD, free-carrier dispersion.

Free-carrier Drude slow-light solitonH Zhou et al

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Light: Science & Applicationsdoi:10.1038/lsa.2017.8

kinetics. Our high-fidelity measurements and analytical solutionscaptured the self-induced soliton acceleration, the self-frequencysoliton blue-shift and the asymmetric spectral broadening under780 fs pulse excitations. We further extended the Drude plasmaperturbation model to broadband MI, illustrating a thresholdless MIgain spanning across the regions of normal and anomalous GVDs. TheDrude dynamics also modified the resonance condition for Cherenkovradiation. To uncover the stability limits of the Drude dynamics, wesubsequently examined the real-time and single-shot soliton-to-solitondynamics through the ADFT. With the Drude plasma, the solitonhistograms illustrate near-Gaussian limited distributions with pumpand spectral sideband cross-correlations at the instrumentation noiselimits. These observations advance fundamental insights into thecombinational Drude plasma and Kerr dynamics of picojoule solitonsin chip-scale silicon dispersive media.

CONFLICT OF INTERESTThe authors declare no conflict of interest.

ACKNOWLEDGEMENTS

We acknowledge discussions with and contributions by Pierre Colman, BahramJalali, Stefano Trillo, Jiangjun Zheng, Tingyi Gu, Jinghui Yang, Hao Zhou,Richard M Osgood Jr, Chad A Husko, Matthew D Marko, Pin-Chun Hsieh,Jiali Liao and Jiankun Yang. Funding support is from the Office of NavalResearch with grant N00014-14-1-0041, and HZ acknowledges UESTC YoungFaculty Award ZYGX2015KYQD051 and the 111 project (B14039). XLacknowledges funding from NSFC Grant 61070040. SWH acknowledgesfunding from AFOSR Young Investigator Award with grant FA9550-15-1-0081.

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Free-carrier Drude slow-light solitonH Zhou et al

8

Light: Science & Applications doi:10.1038/lsa.2017.8

S-1

Supplementary Information

Real-time dynamics and cross-correlation gating spectroscopy of free-carrier Drude slow-light

solitons

H. Zhou1,4,*,†

, S.-W. Huang2,4,*,‡

, X. Li3,4

, J. F. McMillan4, C. Zhang

5, K. K. Y. Wong

5, M. Yu

6,

G.-Q. Lo6, D.-L. Kwong

6, K. Qiu

1, and C. W. Wong

2,4,§

1 Key Lab of Optical Fiber Sensing and Communication Networks, University of Electronic Science

and Technology of China, Chengdu 611731, China. 2 Mesoscopic Optics and Quantum Electronics Laboratory, University of California, Los Angeles,

CA 90095, USA. 3 College of Science, National University of Defense Technology, Changsha, Hunan 410073, China.

4 Optical Nanostructures Laboratory, Columbia University, New York, NY 10027, USA.

5 Department of Electrical and Electronic Engineering, The University of Hong Kong, Pokfulam

Road, Hong Kong. 6 Institute of Microelectronics, Agency for Science, Technology and Research, Science Park II,

117685, Singapore.

*equal contribution. E-mail: † [email protected]; ‡ [email protected];

§ [email protected].

I. Soliton perturbation under the influence of Drude free carrier dispersion

Here we conduct soliton perturbative analysis under the two-photon absorption induced Drude

free carrier dispersion (FCD). Assuming the pulse width is much shorter than carrier lifetime and

neglecting the propagation loss, the NLSE can be simplified and normalized by introducing the

following dimensionless variables

𝑢 = √𝛾𝑒𝑓𝑓𝐿𝐷𝐸 ; 𝜉 =𝑧

𝐿𝐷 ; 𝜏 =

𝑡

𝑇0 (𝑆1)

, where 𝐿𝐷 =𝑇0

2

|𝛽2| is the dispersion distance and 𝑇0 is the pulse width.

The normalized NLSE can be obtained after some algebra as

𝑖𝜕𝑢

𝜕𝜉+

1

2

𝜕2𝑢

𝜕𝜏2+ |𝑢|2𝑢 = 𝜅𝑢 ∫ |𝑢|4𝑑𝑡′

𝜏

−∞

(𝑆2)

, where 𝜅 =𝑘0𝑘𝑐|𝛽2|𝛽𝑇𝑃𝐴

2ℎ𝜈0𝐴02𝛾𝑒𝑓𝑓

2 𝑇0 is the dimensionless Drude FCD coefficient. Treating the FCD term as a

perturbation (the assumption holds with limited pulse energies), the perturbed soliton solution of the

NLSE can then be written as [SR1]

S-2

𝑢(𝜉, 𝜏) = 𝐴𝑚𝑠𝑒𝑐ℎ(𝐴𝑚(𝜏 − 𝑞))𝑒𝑥𝑝(−𝑖Ω𝜏) (𝑆3)

where Am is the perturbed amplitude, q is the self-temporal shift, and is the self-frequency shift.

Using the variational method based on the Euler–Lagrange equation [SR1, SR2], the evolution of

soliton parameters as functions of propagation can be determined by the following equations:

𝑑𝐴𝑚

𝑑𝜉= 𝑅𝑒 {∫ (−𝑖𝜅|𝑢|2 ∫ |𝑢|4𝑑𝑡′

𝜏

−∞

) 𝑑𝜏∞

−∞

} (𝑆4)

𝑑Ω

𝑑𝜉= −𝐼𝑚 {∫ 𝑡𝑎𝑛ℎ[𝐴𝑚(𝜏 − 𝑞)] (−𝑖𝜅|𝑢|2 ∫ |𝑢|4𝑑𝑡′

𝜏

−∞

) 𝑑𝜏∞

−∞

} (𝑆5)

𝑑𝑞

𝑑𝜉= −Ω +

1

𝐴𝑚𝑅𝑒 {∫ (𝜏 − 𝑞) (−𝑖𝜅|𝑢|2 ∫ |𝑢|4𝑑𝑡′

𝜏

−∞

) 𝑑𝜏∞

−∞

} (𝑆6)

As the integral in Eq. S4 is a pure imaginary number, we obtain the result that the soliton amplitude

is not influenced by the Drude FCD. Without loss of generality, we set Am = 1 and the right hand side

of Eqs. S5-S6 can be integrated via Matlab symbolic algebra kit:

𝑑Ω

𝑑𝜉=

8

15𝜅 (𝑆7)

𝑑𝑞

𝑑𝜉= −Ω (𝑆8)

The evolution of soliton self-frequency and temporal shift can be solved as:

∆𝜔 =Ω

𝑇0=

4

15

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴|𝛽2|2

ℎ𝜈0𝐴02𝛾𝑒𝑓𝑓

2 𝑇04 𝑧 =

4

15

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴

ℎ𝜈0𝐴02 𝑧 (𝑆9)

∆𝑡 = 𝑞𝑇0 = −2

15

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴|𝛽2|3

ℎ𝜈0𝐴02𝛾𝑒𝑓𝑓

2 𝑇04 𝑧2 = −

2

15

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴|𝛽2|

ℎ𝜈0𝐴02 𝑧2 (𝑆10)

On the right hand side of Eqs. S9 and S10, the fundamental soliton condition 𝑁2 =𝛾𝑒𝑓𝑓𝑇0

2𝑃0

|𝛽2|= 1 is

applied.

II. Effect of linear loss on pulse propagation and spectrum broadening

The effect of linear loss on self-phase modulation is examined theoretically earlier in Ref. [SR3].

Figure S1 below shows the pulse propagation for different propagation along the photonic crystal for

different linear losses: 10 dB/cm, 25 dB/cm, and 50 dB/cm. It is observed that linear loss does

influence the pulse evolution in the silicon waveguide. In particular, as illustrated in Figure S1c

S-3

a b c

Figure S1 | Pulse propagating along photonic crystal for different linear losses. a, Evolution of

the 1555 nm and 7.5 pJ input pulse; b, Pulse energy as a function of propagation distance. c, Output

pulse waveforms and spectra under different waveguide losses.

below, for the 7.5 pJ pulse at 1555 nm, the spectrum-peak wavelength (temporal peak position) shifts

1.31 nm (-2.6 ps), 1.05 nm (-2.1 ps), and 0.94 nm (-1.6 ps) for 10 dB/cm, 25 dB/cm (the real device),

50 dB/cm loss respectively. In our measurements, the soliton number N is kept above 1, allowing the

path-averaged solitons to broaden less [SR4].

Moreover, photonic crystal waveguides exhibit wavelength-dependent linear loss due to

photonic bandgaps. Such limited transmission can influence the spectrum broadening of femtosecond

pulses, as we discussed in Figure 4 of the main text. As shown in Figure S2, the measured

transmission drops off fast at 1520 nm and 1566 nm. Correspondingly, the broadened pulse spectrum

is subject to spectral filtering. This is confirmed by comparing the experimental measured spectrum

(solid black line) and NLSE simulated spectrum without embedding the wavelength- dependent

linear loss (dashed black line), as illustrated in Figure S2.

In addition, we also noticed that such wavelength-dependent linear loss would affect the

accuracy of our discussion about Cherenkov radiation in Figure 4b of the main text. As exhibited in

Figure S2, with the measured transmission included in NLSE, the Cherenkov radiation sideband is

partially truncated and suppressed (solid green line, with 0% FCD), while the intact Cherenkov

spectrum is notably higher and broader (dashed green line). Therefore, in Figure 4b of our main text,

we replace the dispersive linear loss with a constant loss factor (13.0 dB/mm) so as to warrant

accurate analysis on the FCD perturbation and resonant condition of Cherenkov radiation.

S-4

No

rm.

sp

ectr

um

(d

B)

Lin

ea

r lo

ss(d

B)

1520 1540 1560-80

-60

-40

-20

0

-40

-30

-20

-10

Wavelength (nm)

Figure S2 | Influence of wavelength-dependent linear loss of photonic crystal waveguide on

femtosecond pulse spectrum broadening and Cherenkov radiation. The dashed red line to the right-Y

axis shows the measured linear loss curve. The dashed black line shows the NLSE simulated

broadened spectrum of 780 fs pulse without including the measured transmission curve, and the solid

black line shows the experimental measured spectrum, illustrating clearly the impact of linear loss.

Solid (dashed) green line shows the NLSE simulated Cherenkov radiation spectrum with (without)

including the measured dispersive transmission, showing the effect of linear loss on the analysis of

Cherenkov radiation.

III. Low dispersion-velocity dispersion waveguide via symmetric lattice shift

The photonic crystal waveguide used for Figure 4 in the main text is with specifically designed

low group-velocity dispersion (GVD) regime via symmetric shift of air-holes in the third row apart

from the single line defect. Such lattice shift can significantly modify the band structure of the planar

photonic crystals, giving rise to the flexible group index and GVD of the defect-guided modes

[SR5-SR7]. Figure S3a shows the optical micrograph of the chipset we used which contains groups

of waveguide with a multitude of air-hole shift s (from 0 to 0.25a, a is the lattice constant). Using

plane-wave expansion, we solved for the band structure and dispersion of the sample measured for

Figure 4a (Group 2, Set 3, and Length 1, as marked in Figure S3a). The calculated photonic band

structure is shown in Figure S3b, with the ninth band around 1550 nm (our measurement

wavelengths). The inset of Figure S3b shows the resolved ng and GVD curve from the ninth band. It

S-5

Wavelength (nm)

GV

D (

ps

2/m

m)

Band used around 1550 nm

Wavelength (nm)

ng

GV

D (p

s2/m

m)

a = 404 nm

1520 1530 1540 1550 15600

25

50

-20

0

20

a

b

c

a: 404 nm; r = 0.255a;

s = 0.25a; h = 0.60a

1520 1530 1540 1550 1560-20

0

20

PWE simulation

Polynomial fitting of PWE simulation

(GVD used in Figure 4)

Group 2, Set 3

Figure S3 | Chip micrograph and GVD of the lattice-shifted low-GVD photonic crystal

waveguide. a, Optical micrograph containing the low-GVD photonic crystal waveguides. b,

Computed band structure of photonic crystal waveguide tested, through plane-wave expansion, in

Figure 4 of the main text. The inset shows the corresponding group index and GVD curves resolved

from the ninth band around 1550 nm, with lattice constant a = 404 nm. c, Comparison between the

plane-wave-expansion-simulated GVD versus the fourth degree polynomial fitted GVD used in

Figure 4 of the main text, with β2 = −3.5 ps2/mm, β3 = 0.5 ps

3/mm, β4 = −0.005 ps

4/mm, and β5 =

−0.0025 ps5/mm, centered at 1560 nm. Inset shows the schematic of symmetric third-row shift (s) of

the air-holes and parameters used in plane-wave expansion calculation.

is observed that the increase of ng as a function of wavelength is reduced within 1530 nm to 1545 nm,

corresponding a low-GVD section therein and a zero crossing near 1545 nm [SR5]. To apply the

simulated GVD curve in NLSE calculation, we conducted fourth-order polynomial fitting, with β2 =

−3.5 ps2/mm, β3 = 0.5 ps

3/mm, β4 = −0.005 ps

4/mm, and β5 = −0.0025 ps

5/mm, centered at 1560

nm. The comparison between simulated and polynomial fitted GVD are displayed in Figure S3c,

exhibiting good consistency.

S-6

IV. Modulation instability under the influence of TPA free carrier dispersion:

The electron-hole Drude dynamics also has concurrent effects on the modulation instability and

spectral broadening. Here we analytically derived the MI gain spectrum according to the NLSE

model with the Drude free-carrier dynamics shown earlier in Eq. 2. Using linear stability analysis,

we assume

𝐸 = [√𝑃 + 𝑎(𝑧, 𝑡)]𝑒𝑖[𝛾𝑒𝑓𝑓𝑃−

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴𝜏𝑓

2ℎ𝜈0𝐴02 𝑃2]𝑧

(𝑆12)

𝑁𝑐 =𝛽𝑇𝑃𝐴𝜏𝑓

2ℎ𝜈0𝐴02 𝑃2 + 𝑏(𝑧, 𝑡) (𝑆13)

Here a and b are the perturbations. Here the implicit assumption is the absorption is not yet too

strong such that we can assume the steady-state solution has constant amplitude along the

propagation. This is a necessary approximation because no analytic steady-state solution exists if we

consider all the dynamical absorptions. It is valid at the beginning of the propagation where the MI

starts to grow. The linearized equations are:

𝑖𝜕𝑎

𝜕𝑧−

𝛽2

2

𝜕2𝑎

𝜕𝑡2+ 𝛾𝑒𝑓𝑓𝑃(𝑎 + 𝑎∗) − 𝑘0𝑘𝑐√𝑃𝑏 = 0 (𝑆14)

𝜕𝑏

𝜕𝑡=

𝛽𝑇𝑃𝐴𝑃3/2

ℎ𝜈0𝐴02

(𝑎 + 𝑎∗) −𝑏

𝜏𝑓 (𝑆15)

Now assume:

𝑎(𝑧, 𝑡) = 𝑎1𝑒𝑖(𝐾𝑧−Ω𝑡) + 𝑎2𝑒−𝑖(𝐾𝑧−Ω𝑡) (𝑆16)

𝑏(𝑧, 𝑡) = 𝑏0𝑒𝑖(𝐾𝑧−Ω𝑡) + 𝑏0∗𝑒−𝑖(𝐾𝑧−Ω𝑡) (𝑆17)

Plugging into the linearized equations, and collecting terms containing 𝑒𝑖(𝐾𝑧−Ω𝑡) and 𝑒−𝑖(𝐾𝑧−Ω𝑡),

then we have the following three equations:

𝑏0 =𝛽𝑇𝑃𝐴𝑃3/2

ℎ𝜈0𝐴02 (

1𝜏𝑓

− 𝑖Ω)(𝑎1 + 𝑎2

∗) (𝑆18)

0 = −𝐾𝑎1 +𝛽2Ω2

2𝑎1 + 𝛾𝑒𝑓𝑓𝑃𝑎1 + 𝛾𝑒𝑓𝑓𝑃𝑎2

∗ − 𝑘0𝑘𝑐√𝑃𝑏0 (𝑆19)

0 = 𝐾𝑎2 +𝛽2Ω2

2𝑎2 + 𝛾𝑒𝑓𝑓𝑃𝑎2 + 𝛾𝑒𝑓𝑓𝑃𝑎1

∗ − 𝑘0𝑘𝑐√𝑃𝑏0 (𝑆20)

Replacing b0 by a1 and a2*, we get two equations:

[𝑀11 𝑀12

𝑀21 𝑀22] (

𝑎1

𝑎2∗) = 0 (𝑆21)

S-7

where: 𝑀11 = −𝐾 +𝛽2Ω2

2+ 𝛾0𝑃 −

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴𝑃2

ℎ𝜈0𝐴02(

1

𝜏𝑓−𝑖Ω)

, 𝑀12 = 𝑀21 = 𝛾0𝑃 −𝑘0𝑘𝑐𝛽𝑇𝑃𝐴𝑃2

ℎ𝜈0𝐴02(

1

𝜏𝑓−𝑖Ω)

, 𝑀22 = 𝐾 +

𝛽2Ω2

2+ 𝛾0𝑃 −

𝑘0𝑘𝑐𝛽𝑇𝑃𝐴𝑃2

ℎ𝜈0𝐴02(

1

𝜏𝑓−𝑖Ω)

. Nontrivial solution exists only if 𝑀11𝑀22 − 𝑀12𝑀21 = 0, then we have:

𝐾 = ± |𝛽2Ω

2| √Ω2 +

4𝛾𝑒𝑓𝑓𝑃

𝛽2−

4𝑘0𝑘𝑐𝛽𝑇𝑃𝐴𝑃2

ℎ𝜈0𝐴02𝛽2 (

1𝜏𝑓

− 𝑖Ω) . (𝑆22)

The resulting MI gain is found from 2·Im{K} and shown in Figure 6 of the main text. Figure S4

compares the MI gain spectra with and without the Drude FCD effect, calculated using Eq. S22. It is

seen that, different from conventional Kerr MI utilized in optical parametric oscillators, MI gain is

always guaranteed by Drude FCD regardless of MI frequency and present in both normal and

anomalous group velocity dispersion.

Figure S4 | Comparison of MI gain spectra analytically calculated with (black) and without (red) the

Drude FCD effect. While the canonical Kerr MI has an apparent pump threshold, the FCD-perturbed

MI features a thresholdless amplification and a broader gain bandwidth.

Supplementary References:

[SR1] G. P. Agrawal, Nonlinear Fiber Optics, Third Edition , Academic Press San Diego, 2001.

[SR2] M. F. Saleh, W. Chang, P. Hoelzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and

F. Biancalana, Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled

hollow-core photonic crystal fibers. Phys. Rev. Lett. 107, 203902 (2011).

[SR3] C. Husko, P. Colman, S. Combrié, A. De Rossi, and C. W. Wong, Effect of multiphoton

S-8

absorption and free carriers in slow-light photonic crystal waveguides, Opt. Lett. 36, 2239 (2011).

[SR4] L. Yin, Q. Lin, and G. P. Agrawal, Dispersion tailoring and soliton propagation in silicon

waveguides. Opt. Lett. 31, 1295 (2006).

[SR5] Y. Hamachi, S. Kubo, and T. Baba, Slow light with low dispersion and nonlinear

enhancement in a lattice-shifted photonic crystal waveguide, Opt. Lett. 34, 1072 (2009).

[SR6] S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni and T. F. Krauss, Dispersion

engineered slow light inphotonic crystals: a comparison. Journal of Optics 12, 101004 (2010).

[SR7] P. Colman, S. Combrié, G. Lehoucq, and A. De Rossi, Control of dispersion in photonic

crystal waveguides using group symmetry theory, Opt. Exp. 20, 13108 (2012).