Supercontinuum Generation by Saturated χ^{(2)} Interactions
Marc Jankowski, Carsten Langrock, Boris Desiatov, Marko Loncar, M.M. Fejer
11 Supercontinuum Generation by Saturated χ ( ) Interactions M ARC J ANKOWSKI , C
ARSTEN L ANGROCK , B ORIS D ESIATOV , M ARKO L ON ˇCAR , AND
M. M.F
EJER NTT Research Inc. Physics and Informatics Labs, 940 Stewart Drive, Sunnyvale, California Edward L. Ginzton Laboratory, Stanford University, Stanford, California, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts * Corresponding author: [email protected] February 26, 2021
We demonstrate a new approach to supercontinuum generation and carrier-envelope-offset detection indispersion-engineered nanophotonic waveguides based on saturated second-harmonic generation of fem-tosecond pulses. In contrast with traditional approaches based on self-phase modulation, this techniquesimultaneously broadens both harmonics by generating rapid amplitude modulations of the field en-velopes. The generated supercontinuum produces coherent carrier-envelope-offset beatnotes in the over-lap region that remain in phase across 100’s of nanometers of bandwidth while requiring <
10 picojoulesof pulse energy. © 2021 Optical Society of America http://dx.doi.org/10.1364/ao.XX.XXXXXX
INTRODUCTION
The generation of coherent supercontinua from mode-lockedlasers is an increasingly important nonlinear process found inmany modern optical systems[1–3]. Traditional approaches tosupercontinuum generation (SCG) rely on self-phase modula-tion (SPM) due to χ ( ) interactions in highly nonlinear fibersor in nanophotonic waveguides, and typically require pulseenergies on the order of 100 - 1000 pJ[4]. When used for carrier-envelope-offset (CEO) detection, these devices are followed bya discrete second-harmonic generation (SHG) stage, which fre-quency doubles the long-wavelength portion of the supercon-tinuum to overlap with shorter wavelengths and generate an f -2 f beatnote. Recent work has focused on simplifying the oper-ation of these systems, both by reducing the energy required toachieve octave-spanning SCG and by integrating the SHG stageinto the same nonlinear waveguide. Energy reductions are typi-cally achieved by fabricating long waveguides that utilize the P ∝ L − scaling of the necessary power associated with χ ( ) de-vices, where L is the length of the nonlinear waveguide; the cur-rent state-of-the-art devices operate with 10’s of pJ[5, 6]. Deviceswith integrated SHG stages typically use parasitic χ ( ) processesto achieve frequency doubling during spectral broadening[7–11].This allows for f -2 f beatnotes to be detected from the outputof a single nonlinear photonic device. All of these approachesto f -2 f detection typically produce a narrowband second har-monic, which requires the f -2 f beatnotes to be filtered down toa ∼ χ ( ) processes entirely, and demonstrated the gen-eration of several octaves of bandwidth using 10-pJ of pulseenergy[12]. However, to date, the mechanisms responsible forspectral broadening in this system have been poorly understood.We address these questions in this article.We describe here a new approach to SCG and CEO detectionbased on saturated second-harmonic generation in a dispersion-engineered thin-film PPLN waveguide. In contrast to traditionalapproaches based on SPM, this process generates coherent oc-taves of bandwidth simultaneously for both the fundamentaland second harmonic by introducing rapid amplitude modula-tions onto the pulse envelopes associated with each harmonic.Saturated SHG enables the generation of coherent octaves ofbandwidth with substantially lower energy requirements thanprocesses based on χ ( ) nonlinearities due to both the relativestrength and the P ∝ L − power scaling of χ ( ) processes. Fur-thermore, devices based on this process can generate f -2 f beat-notes in the region where the spectra associated with each har-monic overlap. Remarkably, these beatnotes can remain in phaseacross 100’s of nanometers of bandwidth, which enables efficientCEO detection without the need for narrowband filters. Thephysical processes studied here clarify the behavior of the de-vices studied in [12], and provide a set of design rules for SCGdevices based on saturated χ ( ) nonlinearities.This paper proceeds in four sections: 1,2) We develop an a r X i v : . [ phy s i c s . op ti c s ] F e b -6 -4 -2 0 2 4 6Phase (Radians) -200 -100 0 100 200Time (fs)0123456 P o s i t i on ( mm ) FundamentalEnvelope -100 0 100 200Time (fs)Second HarmonicEnvelope-200 -100 0 100 200Time (fs)0123456 P o s i t i on ( mm ) FundamentalPhase -100 0 100 200Time (fs)Second HarmonicPhase a) b)c) d)
Fig. 1. a,b) Theoretical evolution of the fundamental and sec-ond harmonic envelopes, | A ω | and | A ω | . Dashed whitelines: conversion half-periods during propagation, dashedblack lines: in-coupled fundamental pulse, solid black lines:the resulting pulse at the output of the waveguide for eachharmonic. c,d ) The phase of the fundamental and second har-monic. The fundamental forms plateaus of constant phase,and the second harmonic exhibits a phase that is indepen-dent of time. Spectral broadening is predominantly due tothe femtosecond-scale amplitude modulations on each pulseenvelope, which can generate coherent octaves of bandwidth.analytic model of SCG by saturated SHG in the time and fre-quency domain, respectively, and verify this model using split-step Fourier methods. We then calculate the f -2 f beatnotes in theregion of spectral overlap between the two harmonics and showthat these beatnotes can remain in phase across broad band-widths. 3) We summarize the design of the nonlinear waveg-uides under study and present experimental results, which ex-hibit good agreement with the theoretical approach establishedin Sec. 1-2. When driven with 50-fs-long pulses from a 2- µ moptical parametric oscillator (OPO), these waveguides generatecoherent octaves of bandwidth with as little as 4 pJ of pulse en-ergy coupled into the waveguide. 4) We discuss the scalability ofthis approach, both to lower pulse energies and to longer pulses.This approach to SCG exhibits favorable scaling laws comparedto devices based on SPM, and may potentially realize SCG with100’s of femtojoules in cm-scale devices. However, the energyrequirements to achieve an octave of bandwidth scale with thecube of the input pulse duration. Therefore, SCG and f -2 f de-tection using saturated SHG in nanophotonic PPLN waveguidesis only practical when the input pulse duration is below 100 fs.
1. TIME-DOMAIN THEORY
We consider the evolution of phase-mismatched fundamen-tal and second-harmonic pulses in a dispersion-engineerednanophotonic PPLN waveguide in the limit where the fieldis sufficiently intense to deplete the fundamental and dispersion is negligible over the bandwidth of the pulses. This quasi-staticmodel is motivated by several observations[12]: i) previousexperimental demonstrations have shown that the generatedsupercontinua maintain coherence for soliton numbers far inexcess of χ ( ) devices, which suggests that the spectral broad-ening mechanisms may be different than an effective χ ( ) dueto cascaded χ ( ) interactions [13–15], ii) the observed relativeintensity of the second harmonic violates the assumption of anundepleted fundamental associated with the cascade regime,and iii) the amount of group-velocity dispersion, k (cid:48)(cid:48) ω , at the fun-damental is too small to cause significant self-compression.The coupled-wave equations for the complex field envelopes A ω ( z , t ) and A ω ( z , t ) are given by ∂ z A ω = ˆ D ω A ω − i κ A ω A ∗ ω exp ( − i ∆ kz ) , (1a) ∂ z A ω = − ∆ k (cid:48) A ω + ˆ D ω A ω − i κ A ω exp ( i ∆ kz ) , (1b) where A ω is normalized such that | A ω | is the instanta-neous power of the fundamental wave. The temporalwalk-off ∆ k (cid:48) = k (cid:48) ω − k (cid:48) ω and dispersion operators ˆ D ω = ∑ ∞ j = (cid:104) ( − i ) j + k ( j ) ω / j ! (cid:105) ∂ jt , where k ( j ) ω represents the j th deriva-tive of propagation constant k at angular frequency ω , areboth assumed to be negligible in the quasi-static limit treatedhere. ∆ k = k ω − k ω − π / Λ G is the phase mismatch be-tween the carrier frequencies of the interacting harmonics inthe presence of a PPLN grating with period Λ G , and κ = (cid:113) ω d / (cid:0) n ω n ω (cid:101) c A eff (cid:1) = √ η is the nonlinear coupling,where η is the conventional normalized SHG efficiency. Inthis quasi-static limit, Eqns.1a-1b may be solved for the instan-taneous field intensity at each point in time using the Jacobi-Elliptic functions associated with continuous-wave SHG in thelimit of a depleted pump[16]. The local instantaneous power ofthe fundamental and second harmonic are given by | A ω ( z , t ) | =( − η ( z , t )) | A ω ( t ) | , (2a) | A ω ( z , t ) | = η ( z , t ) | A ω ( t ) | , (2b) where the instantaneous field conversion efficiency is givenby η ( z , t ) = ν ( t ) sn (cid:0) κ A ω ( t ) z / ν ( t ) | ν ( t ) (cid:1) . Here sn isthe Jacobi elliptic sine and ν ( t ) = −| ∆ k / ( κ A ω ( t )) | + (cid:112) + | ∆ k / ( κ A ω ( t )) | . ν ( t ) represents the maximum pumpdepletion attainable as a function of the local field amplitude A ω ( t ) .The Jacobi elliptic solutions found here bear many similari-ties to the sine-wave evolution that occurs during undepletedphase-mismatched SHG, namely, periodic oscillations of the fun-damental and second harmonic power in z , and a maximumconversion efficiency that increases with increasing pump power.However, the Jacobi-elliptic functions saturate at high power,and the spatial period at which power oscillates between thefundamental and second harmonic, hereafter referred to as theconversion period, decreases as the local field intensity becomeslarger. The conversion period varies across the field envelopesas L conv ( t ) = K ( ν ( t ) ) ν ( t ) / ( κ A ω ( t )) , (3) where K is the complete elliptic integral of the first kind. K ( ν ( t ) ) varies slowly for most physically encountered values of ν ( t ) ,e.g. K ( ) = π /2 and K ( ν ) ≈ ν = L conv ( t ) is dominatedby ν ( t ) / ( κ A ω ( t )) . Fig. 1(a-b) shows the theoretical evolu-tion of a 50-fs-wide sech pulse in a 6-mm-long waveguide. -200 -100 0 100 200Time (fs)0123456 P o s i t i on ( mm ) TheoryFundamental -100 0 100 200Time (fs)TheorySecond Harmonic-200 -100 0 100 200Time (fs)0123456 P o s i t i on ( mm ) SimulationFundamental -100 0 100 200Time (fs)SimulationSecond Harmonica) b)c) d)
Fig. 2.
Comparison of quasi-static theory with a full split-stepFourier simulation. a,b) Theoretical fundamental and secondharmonic, c,d) simulated fundamental and second harmonic.The structure of the amplitude-modulated harmonics is largelypreserved in the presence of moderate amounts of waveguidedispersion.Here, we have assumed a pulse energy of 4 pJ, η = , and ∆ k = − π / L , where L is the length of the waveg-uide. The dotted white lines correspond to the m th half-period, L m = mL conv /2, where even m coincide with the local max-ima and minima of the fundamental and second harmonic, re-spectively. Near the peak of the pulse the conversion periodis the shortest and both harmonics undergo ∼ L conv ( ∞ ) = π / | ∆ k | . Remarkably, the power at the peakoscillates three times faster than in the tails of the pulse, whichgives rise to a pulse shape with rapid temporal amplitude os-cillations as each portion of the pulse cycles through a differentnumber of conversion periods (Fig. 1, solid lines).Using the same quasi-static heuristic, the Jacobi elliptic so-lutions can be shown to predict phase envelopes for the funda-mental and second harmonic, φ ω ( z , t ) = φ ω ( t ) + ∆ k (cid:90) z η ( z (cid:48) , t ) − η ( z (cid:48) , t ) dz (cid:48) , (4a) φ ω ( z , t ) = φ ω ( t ) − π /2 + ∆ k z , (4b) respectively, and are plotted in Fig. 1(c,d) for φ ω ( t ) =
0. Therate of phase accumulation by the fundamental depends stronglyon the degree of pump depletion, with large phase shifts accu-mulated at values of z and t that correspond to local maximaof η ( z , t ) . This behavior results in a saturable effective SPM forthe fundamental, with the total accumulated phase plateauingacross large time bins (Fig. 1(c)). The phase of the second har-monic is independent of time, and therefore can be neglectedin the context of spectral broadening (Fig. 1(d)). These twobehaviors suggest that the predominant broadening mechanismfor saturated SHG is not effective SPM. Instead, the observed
50 100 150 200 250 300 350Frequency (THz)0123456 P r opaga t i on Leng t h ( mm ) -60 -50 -40 -30 -20 -10 0Power Spectral Density (dB)a) Theoretical Beatnote Current (Arb. Units)50 100 150 200 250 300 350Frequency (THz)456 P r opaga t i on Leng t h ( mm ) -300 -200 -100 0 100 200 300b) Propagation Length (mm) B ea t no t e C u rr en t ( A r b . U n i t s ) c) Fig. 3. a) Theoretical power spectral density associated withthe field envelopes shown in Fig. 1. b) Beatnote photocurrentas a function of frequency and propagation length. The beat-notes periodically fall in- and out-of-phase with propagationlength, with some lengths producing beatnotes that remainin-phase across 100’s of nm of bandwidth. c) The resultingdetected photocurrent oscillates for increasing propagationlength due to the periodic re-phasing shown in (b).spectral broadening of the harmonics is dominated by the rapidamplitude modulations in time accumulated by each pulse.We now verify this analytic model against a full split-stepFourier model, assuming a temporal walk-off of ∆ k (cid:48) = k (cid:48)(cid:48) ω = -15 fs /mm and k (cid:48)(cid:48) ω =
100 fs /mm, respec-tively. These values were chosen to correspond to the temporalwalk-off and group-velocity dispersion of the TE waveguidemodes of the waveguides studied in Sec. III. The time-domain in-stantaneous power, | A ω | and | A ω | , is shown in Fig. 2. Whilethe simulated pulse envelopes exhibit some distortion due tosecond and third-order dispersion, the key aspects of our Jacobi-elliptic approach such as the rapid amplitude modulations ofthe resulting pulses are largely preserved. Given this strongagreement, we now consider the spectral broadening of the har-monics.
2. FREQUENCY-DOMAIN THEORY
Having calculated both the amplitude and phase of each enve-lope, we may now Fourier transform these envelopes to studythe evolution of the power spectral density. While we cannotFourier transform the fields obtained using Eqns. 2 and 4 an-alytically, we may gain several insights from the time-domainmodel that qualitatively capture the behavior of the generatedspectrum. First, as noted previously, the phase of the two har-monics has a negligible contribution to spectral broadening, with φ ω forming plateaus of nearly constant phase and φ ω ( z , t ) = φ ω ( z , 0 ) experiencing no time-dependent phase modulation.Second, we note that for z > L conv ( ) the number of local max-ima contained in the instantaneous power of each envelopegrows linearly with the number of half-periods around t = t → ± ∞ ). Therefore, the number of local maxima foreach harmonic is given by N ω ( z ) ≈ z (cid:2) L − ( ) − L − ( ∞ ) (cid:3) and N ω ( z ) = N ω ( z ) + z > L conv ( ) ; both harmonicshave one local maximum for z < L conv ( ) . Since the instanta-neous power | A ω ( z , t ) | + | A ω ( z , t ) | is conserved, each fieldenvelope effectively bifurcates into N pulses with a duration ∼ τ / N , where τ is the pulse duration input to the waveguide(e.g. τ FWHM = τ for a sech pulse). Based on these observa-tions we expect three behaviors in the frequency domain: i) abandwidth ∆ f that grows linearly for z that satisfy N ω ( z ) > z (cid:29) L conv ( ) ), ii) a constant bandwidth for z that satisfy N ω ( z ) <
2, and iii) the appearance of fringes in the frequencydomain that become more finely patterned with increasing z ,due to the interference of N pulses in the time-domain.The combined power spectral density associated with eachharmonic, | ˆ A ω ( z , Ω ) | + | ˆ A ω ( z , Ω ) | is plotted in Fig. 3(a), forthe parameters used in Fig. 1. The white dotted lines correspondto a semi-empirical formula for the spectral half-width based onthe arguments made above, ∆ f ω ( z ) = ∆ f ( ) (cid:113) + ( ¯ N ω ( z ) /2 ) , (5) where ¯ N ω = z (cid:104) z − − L − ( ∞ ) (cid:105) , ¯ N ω = ¯ N ω +
1, and z NL = πν ( ) / κ A ω (
0, 0 ) . Here ∆ f ( ) was chosen to correspond to the −
50 dB level. Eqn. 5 is a reasonable estimate of the spectralbroadening due to quasi-static χ ( ) interactions; we numericallyverified that the accuracy of this equation holds as we variedthe pulse energy and phase-mismatch over several orders ofmagnitude. As the field envelopes evolve in the waveguide,the bandwidth associated with each harmonic is observed tobe constant for z (cid:28) z NL , and to grow linearly for z (cid:29) z NL .Furthermore, the power spectral density associated with eachharmonic exhibits interference fringes that become more finelypatterned with increasing z , as expected from the qualitativepicture discussed previously. We note here that the harmonicsmerge for z > f -2 f interferometry in theregion of spectral overlap.The beatnote power contained in each spectral bin is cal-culated using 2Re (cid:0) ˆ A ω ( z , Ω ) ˆ A ∗ ω ( z , Ω ) (cid:1) . Fig. 3(b) shows thecalculated f CEO beatnote current as a function of frequency forthe overlapping spectra. The beatnotes fall in- and out- of phaseduring propagation, and, remarkably, for a suitable choice ofpower or device length the f CEO beatnotes can remain in-phaseacross nearly a micron of bandwidth. As shown in Fig. 3(c),this process causes the beatnote current obtained by integratingover the full bandwidth to oscillate depending on the lengthof the waveguide. In practice, the pulse energy used to drive a) ND OBJ OBJ
Nanophotonic
PPLN50-fs pulses Fiber b) 800 1000 1200 1400 1600 1800 2000 2200 2400246810 P u l s e E ne r g y ( p J ) Power Spectral Density (dBm) −80 −70 −60 −50 −40 −30 −20Experiment
800 1000 1200 1400 1600 1800 2000 2200 2400
Wavelength (nm) P u l s e E ne r g y ( p J ) Theory
Fig. 4. a) Experimental Setup. (OPO) Optical parametric os-cillator, (OBJ) metallic cassegrain objective, (OSA) opticalspectrum analyzer. b) The experimentally measured and, c)theoretically calculated power spectral density as a function ofinput pulse energy.the waveguide can be chosen to align a local maximum of thebeatnote current with the length of the device. This process sim-plifies CEO detection by allowing the output of the waveguideto be focused on a photoreceiver with minimal filtering, whilealso improving the detected beatnote current by integrating thephotocurrent over many comb lines.
3. EXPERIMENTAL RESULTS
Having established the physical processes responsible for spec-tral broadening during saturated SHG, we now experimentallycharacterize SCG in nanophotonic PPLN devices. The designand fabrication of the devices studied here were reported pre-viously in [12, 17, 18], and we summarize the relevant aspectshere. First, we periodically poled a 700-nm-thin film of MgO-doped lithium niobate with with 15 poling periods ranging from5.01 µ m to 5.15 µ m by applying high voltage pulses to elec-trodes patterned on the top surface. This shift of 10 nm between consecutive poling periods corresponds to a shift of the phasemismatch of 4 π . Fine tuning of the phase-mismatch may beachieved by changing the temperature of the waveguide. Sec-ond, waveguides are patterned using electron-beam lithography,and etched into the thin film using Ar + ion-based reactive ionetching. The fabricated waveguides have a top width of ∼ ∼
340 nm, which achieve phasematchedSHG of pulses centered around 2060-nm with a poling period of ∼ µ m.The experimental setup is shown in Fig. 4(a). The waveg-uides are driven using 50-fs-long pulses produced with a repeti-tion rate of 100 MHz from a synchronously pumped degenerateoptical parametric oscillator. These pulses are coupled into thePPLN waveguides using a reflective inverse-cassegrain objective(Thorlabs LMM-40X-P01). This choice ensures that the incoupledpulses are chirp-free and that the collected harmonics are freeof chromatic aberrations. The strongest spectral broadening isobserved for a poling period of 5.10 µ m, such that ∆ kL ∼ π . Werecord the output spectrum from the waveguide using two spec-trometers: the near- to mid-infrared (600-1600 nm) is capturedwith a Yokogawa AQ6370C, and the mid-infrared (1600-2400nm) is captured using a Yokogawa AQ6375. The results areshown in Fig. 4(b). The fundamental and second harmonicare observed to broaden for input pulse energies in excess of100 fJ, with the two harmonic merging at the -40 dB level forpulse energies as low as 4-pJ. This observed broadening withincreasing pulse energy is consistent with the quasi-static theoryshown in Fig. 4(c). Furthermore, we observe a number of quali-tative similarities between the spectra observed in theory andexperiment. In particular, for pulse energies between one to fivepicojoules, the power spectrum of the fundamental exhibits alocal minimum around the carrier frequency of the fundamental,2090 nm. For pulse energies greater than five picojoules, thislocal minimum splits into two minima centered symmetricallyaround the carrier frequency, with a local maximum at 2090 nm.Similar patterns occur in the tails of the spectra; the spectrum ofthe fundamental forms successive local minima and maxima inthe band between 1600 - 1800 nm with increasing pulse energy,and the second harmonic exhibits oscillatory tails between 1200- 1400 nm. However, the experimentally observed second har-monic is much brighter than the theoretical predictions madehere.For energies > f -2 f beatnotes may be detected in theregion of spectral overlap by filtering the light output from thewaveguide and focusing it onto a photodiode (Fig. 5(a)). Fig. 5(b)shows the theoretical beatnote contrast output from the waveg-uide, 2Re (cid:0) ˆ A ω ( L , λ ) ˆ A ∗ ω ( L , λ ) (cid:1) / (cid:0) | ˆ A ω ( L , λ ) | + | ˆ A ω ( L , λ ) | (cid:1) ,as a function of wavelength and pulse energy. The contrastis highest in the spectral region from 1200 - 1800 nm. There-fore, we detect f -2 f beatnotes by filtering the light output fromthe waveguide using a Thorlabs FELH1250 longpass filter, andfocus this light on a MenloSystems APD310 avalanche photo-diode. We record the RF beatnotes using a Rigol DSA815 RFspectrum analyzer with the resolution bandwidth set to 10 kHz.The RF beatnote power in dBm is shown in Fig. 5(c) as a functionof input pulse energy. We observe oscillations of the beatnotepower with increasing pulse energy, which is consistent withperiodic re-phasing of the beatnotes predicted by theory (Fig.5(c), dashed line). The beatnote current achieves a local maxi-mum around an input pulse energy of 9.75 pJ, and we obtain anRF beatnote power of -65 dBm. Fig. 5(d) compares the relativeintensity of the measured f CEO beatnote with the f R beatnotecorresponding to the repetition rate of the pulses for a pulse a) RFND
OBJ OBJ
Nanophotonic
PPLN50-fs pulses APDFilter b) Theoretical Beatnote Contrast
800 1000 1200 1400 1600 1800 2000 2200 2400
Wavelength (nm) P u l s e E ne r g y ( p J ) -1 -0.5 0 0.5 1c) 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Pulse Energy (pJ) -100-90-80-70-60 B ea t no t e P o w e r ( d B m ) d)
50 75 100 125 150 175 200 225 250
Frequency (MHz)−100−90−80−70−60−50−40 B ea t no t e P o w e r ( d B m ) f ceo f R
25 dB
Fig. 5. a) Experimental Setup. (APD) Avalance photodiode,(RF) Radiofrequency spectrum analyzer. b) Theoretical beat-note contrast as a function of wavelength and pulse energy.Based on this calculation, we detect the range from 1250 nm- 1800 nm (dashed grey lines) using a long-pass filter and anInGaAs avalanche photodiode. c) Measured beatnote poweras a function of pulse energy. Orange circles correspond to ex-periment, and the dashed black line corresponds to theory. d)Measured f -2 f beatnotes, for a pulse energy of ∼
10 pJ.energy of 9.75 pJ. The detected f CEO beatenote power is only25 dB below the f R beatnote, even when the detected opticalbandwidth spans ∼
550 nm.
4. SCALABILITY OF THIS APPROACH
In practice, there are two main considerations that determinehow well this technique performs relative to conventional ap- proaches, namely, i) the power requirements and ii) the pulseduration requirements necessary to achieve octaves of band-width. We first consider the energy requirements of SCG bysaturated χ ( ) nonlinearities using Eqn. 1, and compare theresulting scaling laws against traditional approaches to SCGbased on soliton fission in waveguides with χ ( ) nonlineari-ties. Eqns. 1 are scale invariant when z → s z , ∆ k → ∆ k / s , ∆ k (cid:48) → ∆ k (cid:48) / s , ˆ D ω → ˆ D ω / s , ˆ D ω → ˆ D ω / s , and P in ( t ) = | A ω ( t ) | + | A ω ( t ) | → P in ( t ) / s . Therefore, an increaseof device length by a factor s correspondingly results in aquadratic reduction of the pulse energy necessary to achieve thesame degree of spectral broadening at the output, U in → U in / s ,provided that the dispersion of the waveguide remains suffi-ciently negligible over the length of the device. In contrast, thenonlinear Schrödinger equation used to describe SCG in waveg-uides with χ ( ) nonlinearities is scale invariant when z → sz ,ˆ D → ˆ D / s , and P in ( t ) = → P in ( t ) / s , which exhibits a linear re-duction of the energy required to produce a supercontinuum asthe length of the waveguide is increased. While state-of-the-artdevices based on χ ( ) nonlinearities have been able to achieveSCG with 10’s of pJ using long waveguides, the quadratic scal-ing of the energy requirements associated with a χ ( ) processmay enable octave-spanning SCG with 100’s of femtojoules ofpulse energy by rescaling the waveguide designs shown here tocentimeters.We now consider the role of pulse duration by using Eqn. 5.In the limit of large nonlinear coupling ( κ A ω (
0, 0 ) (cid:29) ∆ k ), orlarge z , the bandwidth grows as ∆ f ( z ) = ∆ f ( ) κ A ω (
0, 0 ) z / π .Rescaling the input pulse duration by a factor s reduces boththe input bandwidth and intensity, ∆ f ( ) → ∆ f ( ) / s and A ω (
0, 0 ) → A ω (
0, 0 ) / √ s , such that power required to achievea desired ∆ f increases as U in → U in s . The cubic scaling de-rived here is shown in Fig. 6, where the spectral broadeningof a 4-pJ, 50-fs pulse is compared against a 4-nJ, 500-fs pulse,with each achieving a similar amount of bandwidth at the out-put. This cubic scaling restricts us to pulse durations on theorder of ≤
100 fs simply because the energy requirements oflonger pulses are impractical. Even a 200-fs-long pulse wouldrequire 100’s of picojoules of pulse energy to achieve the outputbandwidths demonstrated here, whereas similar pulse energieshave already been used to produce χ ( ) supercontinua in LNnanowaveguides [10, 19]. While these limitations can be over-come by using longer waveguides or more tightly confining(and therefore more nonlinear) geometries, care must be takento ensure that the nonlinear coupling is sufficient to produceoctaves of bandwidth when using pulses with a longer duration.
5. CONCLUSION
We have established a theoretical model of supercontinuum gen-eration based on saturated quasi-static χ ( ) interactions, andhave experimentally verified this model by studying spectralbroadening in PPLN nanowaveguides. In contrast with the ef-fective self-phase modulation that occurs with cascaded χ ( ) interactions, here spectral broadening occurs due to rapid ampli-tude modulations across the pulse envelope of the fundamentaland second harmonic. This process generates coherent octavesof bandwidth with picojoules of pulse energy, and produces f -2 f beatnotes that can remain in-phase across 100’s of nanome-ters of bandwidth. This behavior simplifies f -2 f detection andimproves the signal-to-noise ratio of the detected f CEO beatnotesince the detected photocurrent can be integrated over many -60 -50 -40 -30 -20 -10 0Power Spectral Density (dB)50 100 150 200 250 300 350Frequency (THz)0123456 P r opaga t i on Leng t h ( mm ) a)50 100 150 200 250 300 350Frequency (THz)0123456 P r opaga t i on Leng t h ( mm ) b) Fig. 6.
Theoretical broadening of a) a 50-fs-long, 4-pJ pulse,and b) a 500-fs-long, 4-nJ pulse. Dashed white lines corre-spond to Eqn. 5. For z >> L conv ( ) the spectra grow linearlywith z , and achieve a similar output bandwidth. This cubicscaling of necessary pulse energy with input pulse durationlimits this technique to pulses with a duration ≤
100 fs.comb lines. Finally, we use our model to derive a set of scal-ing laws that provide simple design rules for devices based onsaturated χ ( ) interactions. Funding Information.
National Science Foundation (NSF) (ECCS-1609549, ECCS-1609688, EFMA-1741651); AFOSR MURI (FA9550-14-1-0389); Army Research Laboratory (ARL) (W911NF-15-2-0060, W911NF-18-1-0285).
Acknowledgments.
The authors wish to thank NTT Research fortheir financial and technical support. Electrode patterning and polingwas performed at the Stanford Nanofabrication Facility, the StanfordNano Shared Facilities (NSF award ECCS-2026822), and the Cell SciencesImaging Facility (NCRR award S10RR02557401). Patterning and dryetching was performed at the Harvard University Center for NanoscaleSystems (CNS), a member of the National Nanotechnology CoordinatedInfrastructure (NNCI) supported by the National Science Foundation.The authors thank Jingshi Chi at DISCO HI-TEC America for her exper-tise with laser dicing lithium niobate.
Disclosures.
The authors declare no conflicts of interest.
Data availability.
Data underlying the results presented in this papermay be obtained from the authors upon request.
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