Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves
Logan G. Wright, Stefan Wabnitz, Demetrios N. Christodoulides, Frank W. Wise
UUltrabroadband dispersive radiation by spatiotemporal oscillationof multimode waves
Logan G. Wright, ∗ Stefan Wabnitz, Demetrios N. Christodoulides, and Frank W. Wise School of Applied and Engineering Physics,Cornell University, Ithaca, New York 14853, USA Dipartimento di Ingegneria dell (cid:48)
Informazione,Universit`a degli Studi di Brescia, and Istituto Nazionale di Ottica,CNR, via Branze 38, 25123 Brescia, Italy CREOL, College of Optics and Photonics,University of Central Florida, Orlando, Florida 32816, USA
Abstract
Despite the abundance and importance of three-dimensional systems, relatively little progresshas been made on spatiotemporal nonlinear optical waves compared to time-only or space-only sys-tems. Here we study radiation emitted by three-dimensionally evolving nonlinear optical waves inmultimode fiber. Spatiotemporal oscillations of solitons in the fiber generate multimode dispersivewave sidebands over an ultrabroadband spectral range. This work suggests routes to multipurposesources of coherent electromagnetic waves, with unprecedented wavelength coverage. ∗ [email protected] a r X i v : . [ phy s i c s . op ti c s ] S e p igenmodes are ubiquitous tools for describing complex wave systems. For nonlinear com-plex wave systems, however, the superposition principle is not applicable. In special cases,nonlinear wave systems possess solitons, which act to some extent as nonlinear eigenmodes.Combined with more general nonlinear attractors and perhaps insights from linearized sys-tems, researchers may thus build up a conceptual understanding of complex nonlinear wavedynamics. In optics, one-dimensional (1D) dynamics in single-mode waveguides have beenthoroughly explored, with many advances hinging on the robust nonlinear attraction ofsolitons[1–3]. In unbounded 3D systems, dynamics have been successfully explained largelyin terms of a nonlinearly attracting instability, namely spatial or spatiotemporal collapse[4].In reality, the collapse singularity is avoided by a multitude of higher-order effects, andthe field eventually expands, by propagating linearly. Post-collapse, promising results havebeen obtained considering conical waves, which are the eigenmodes of the 3D linear waveequation[5, 6]. Although not yet widely adopted, conical wave solutions to the nonlin-ear wave equation may provide even deeper insight[7, 8]. Given the great advantages theconcepts of solitons and collapse have provided in the field of nonlinear optical waves forstudying single-mode waveguides and free space filamentation, it is natural to seek simi-larly useful nonlinear objects in multimode waveguides. Multimode waveguides include aslimiting cases single mode (1D) and free-space (3D), so these hypothetical objects, whethersolitons, nonlinearly attracting instabilities, guided conical waves, or something else entirely,could even help to conceptually unify nonlinear optical dynamics across dimensions.Solitons in optical single-mode fiber (SMF) have been intensely researched, because theyare relatively accessible both analytically and experimentally. Equally important, solitondynamics are critical to telecommunications, mode-locked fiber lasers, and compact white-light sources with high spatial mode quality. Multimode fibers (MMFs) could provide majorbenefits for various applications, from spatial division multiplexing in communications[9, 10],to high-power, versatile fiber lasers and white-light sources[11]. Although wave propagationin MMF is still experimentally and theoretically challenging, recent theoretical advances[12–17] combined with opportunities for major performance increases provide ample motivationfor their study. From a scientific perspective, MMF is an ideal environment for studyingspatiotemporal nonlinear optical dynamics. By judicious design, or by control of the initialexcitation, researchers may control the spatiotemporal characteristics of nonlinear dynam-ics, through variation of the effective dimensionality, the coupling between modes, or their2ndividual dispersions.These factors have motivated recent work on nonlinear optical waves in MMFs[11, 15,16, 18–22]. In particular, control of the spatial excitation of a MMF provides a meansof controlling the nonlinear spatiotemporal dynamics, and therefore the characteristics ofa supercontinuum[11]. Launching ≈ >
50 to < µ m). − − I n t en s i t y v ( d B ) Frequencyv(THz)4167 1552 857 750 600 545Wavelengthv(nm) − I n t en s i t y v ( d B ) FIG. 1. Simulated (top) and example experimental (bottom) supercontinuum in multimode GRINfiber. The pump pulse at 1550 nm creates a spectrum with a series of red-shifted and blue-shiftedpeaks. In the bottom panel, the y-axis reference (0 dB) is the maximum intensity of the 1550 nmpump peak.
Here we provide a theoretical explanation for this phenomenon: dispersive waves are gen-erated by the spatiotemporal oscillation of multimode solitons. The process is inherently 3D,with spatiotemporally-evolving nonlinear waves emitting spatiotemporally-evolving disper-3ive waves. However, we show that insights from solitons of the 1D nonlinear Schr¨odingerequation (NSE), can prove useful in understanding the higher-dimensional system. Thisunderstanding also suggests routes to generating ultrashort pulses in fiber at wavelengthsoutside current capabilities, and may provide a means of interfacing normally distinct partsof the electromagnetic spectrum.Solitons are solutions to a conservative equation. To use them in applications such asmode-locked lasers and optical telecommunications[23–34], loss must be compensated. Thisis naturally accomplished by laser gain, or by amplifiers placed throughout the line. Theperiodic perturbations of gain and loss, however, can destabilize a soliton. The origin of theinstability, and the characteristic spectral sidebands that are its signature, is the fact that aperiodic perturbation can phase-match dispersive wave emission at particular frequencies[35,36]. For a perturbation period (spacing of amplifiers, or laser cavity length) Z c , the phase-matching condition is approximately:( k sol − k dis ) = 2 mπ/Z c (1)where m is an integer, and k dis is the wavevector of the dispersive wave. k sol is the soli-ton wavevector, which in the absence of higher-order dispersion is equal to | β | / τ = π/ (4 Z o ),where β is the group velocity dispersion, τ is the soliton duration and Z o = π/ (4 k sol ) is the soliton period. This quasi-phase matching leads to resonant emission atfrequencies separated from the pump byΩ res = 1 τ (cid:114) Z o mZ c − k sol − k dis ) = 2 mπ/Z c = ( 12 + 4 b − b ) − ( − Ω − b Ω ) (3)where b = − β / (6 | β | τ ) ( β is the third-order dispersion) and Ω is the angular frequencyseparation from the pump (in units of τ − ). Production of dispersive waves is the primarylimitation to the performance of soliton fiber lasers.Although they were predicted as early as the 1980s[41–45], solitons consisting of pulseswithin multiple spatial modes have only recently been studied experimentally[11, 18, 22].Initial work shows that solitary waves, termed MM solitons, can be excited in multimode4raded-index (GRIN) fibers. Similarly to single-mode solitons, these pulses resist group-velocity dispersion and adjust themselves adiabatically in response to perturbations such asintrapulse Raman scattering. This behavior makes them a useful conceptual tool for under-standing complex nonlinear processes beyond the NSE description, such as those involved insupercontinuum generation[11, 22]. Unlike single-mode solitons however, MM solitons addi-tionally resist modal velocity dispersion - i.e. , the fact that each of the numerous consitutentmodes of the soliton has a different group velocity. Fission of MM solitons is spatiotem-poral: it yields multiple MM solitons which can have many different modal distributions.In fact, MM solitons can have a bewildering range of spatiotemporal shapes, which makestheir dynamics much richer and challenging than single-mode solitons. While they are insome sense natural extensions of the 1D NSE soliton to the case of a MM fiber, they prob-ably do not fulfill the most rigorous definitions of (cid:48) soliton (cid:48) (some special cases are alreadyknown, however[15, 16]). Questions such as whether they are stable over very long distances,how they interact with one another, and how many modes can be involved, remain largelyunanswered.When a beam excites multiple modes of a GRIN fiber, it propagates through the fiberwith a characteristic spatial oscillation with period (pitch) P = πR/ √ R is thecore radius and ∆ is the core-cladding index difference[18, 46]. This oscillation causes theintensity of the beam to periodically evolve, as in a loss-managed soliton transmission lineor soliton fiber laser. For a pulsed beam, oscillations will only occur as long as the pulses ineach mode maintain co-localization and a phase relationship[18]. Hence, a soliton containingmultiple spatial modes experiences a periodic oscillation of its peak intensity and thereforeis likely to emit dispersive radiation at particular frequencies depending on the period of theoscillation.We consider a simulation using the generalized multimode nonlinear Schr¨odinger equa-tion (GMMNLSE)[12], with the first five radially-symmetric modes excited uniformly forsimplicity (Figure 1). Experiments are conducted with the procedures described in Ref.[11]. In both simulation and experiment, we find that the soliton oscillation-induced dis-persive waves (ODWs) are observed for many different initial spatial conditions, includingnon-radially-symmetric excitations. Typically we observe that the ODWs are more appar-ent when the intensity of the initial pulse is higher (either by large energy, tight spatiallocalization, or both). Changing the initial noise level or the pulse duration results in dif-5 R M S u w i d t h u z µ m
50 100 150 200 250 300 05001000 R M S u du r a t i on u z f s
50 100 150 200 250 30010 Distanceuzperiods9 E ne r g y u z a r b .u un i t s − − − − − − − − − − − − − − − abc I n t en s i t y u z d B
148 149 150 151614 Distanceuzperiods9148 149 150 151200400
FIG. 2. Dynamics of dispersive wave formation in GRIN fiber. a) Temporal and spatial breathingof the field (inset: zoom in near the onset of dispersive wave generation). b) Evolution of thespectral intensity of the whole field through the same distance. c) Energy in each dispersive waveband. The x-axis scales are normalized to the linear spatial oscillation period of the GRIN fiber,equal to 407 µ m. These dynamics are also shown in Supplementary Movies 1 and 2, which providea considerably more complete representation of the complex spatiotemporal evolution. ferent dynamics. Nonetheless, provided the pulse energy and fiber length are sufficient, andthat multiple modes are excited, the qualitative features (including the spectral positions)of the ODW emission process are similar. In simulations, we observe that the energy of eachODW is distributed roughly equally among all the modes, with the red-shifted (blue-shifted)6 I n t en s i t y h d B o b g Ogg +gg fgg 4gg 5gg 6gg − +g − Ogg Frequency5hTHzo m TheoryExperimentSimulation g gWO gW+ gWf gW4 gW5 gW6 gW7 gW8O+f45 z5hmmo γ h z o v γ o
857 75g 667 6gg 545Wavelength5hnmo − − − +gg Frequency5hTHzo − +gg ExperimentalNEEMMGNLSE a MMGNLSELarge5OscillationTypW5Oscillationspace7time5fit
FIG. 3. a. ODW from MMGNLSE simulations (MMGNLSE), eqn. 4 with γ ( z ) as in text,1D NEEsimulations, and an experimental example spectrum. b. Functions used in the quasi-1D approxima-tion, along with the peak intensity variation of the field in MMGNLSE simulation. c.Comparisonof the periodic 1D phase-matching model with multimode simulations by the GMMNLSE and ex-periments in GRIN MMFs. Continuous curves are values of m in Eq. 3, plotted with the best-fitparameter values. sidebands exhibiting a slight preference for the low-order (higher-order) modes.Representative simulation results for a 400-fs, 1550-nm MM pulse with energy 164 nJlaunched into a GRIN MMF are shown in Figure 2. (The results are also shown as Sup-plementary Movies 1 and 2.) The spatial and temporal breathing of the field are evidentin Fig. 2a. While we show only the blue-shifted ODWs (Figure 2b), the simulation alsopredicts red-shifted ODWs. These are remarkably outside the transparency window of fusedsilica (the first appears at roughly 72 THz (4200 nm)). As the pulse traverses the fiber,7he spectrum develops a dispersive wave near 300 THz (1000 nm), as well as ODWs in thevisible and mid-IR regimes. Attenuation is included in the simulations with an assumed fre-quency dependence α = α exp − ( f − c/ (1550 nm )) /b l , where α = 0 . dB/km is theattenuation (units m − ) at 1550 nm, and b l = 0 . ≈
80 nm) models the increasingloss into the infrared. For these parameters, attenuation is ≈ ≈ ∂A ( z, t ) ∂z = − i β ∂ A ( z, t ) ∂t + β ∂ A ( z, t ) ∂t + iγ ( z ) | A ( z, t ) | A ( z, t ) (4)where A ( z, t ) is the pulse envelope, and γ ( z ) is the z -dependent nonlinear coefficient. Fig-ure 3 compares the result of solving this equation to the results above found using theGMMNLSE, and with experiment. Figure 3a shows the solution of equation 4 with theindicated form of γ ( z ). Due to the sinusoidal oscillation of the beam radius, one expects anaccurate γ ( z ) to be of the form γ ( z ) = γ o / ([ r max − r mean ] + r mean sin (2 πz/P )) . We use theRMS widths for r max and r mean here, and rescale γ o appropriately, since the usual spatialwidth measurement (mode field diameter) is not well-defined for the complex spatiotemporalfields that occur. At the onset of the ODW generation ( ≈ r max = 14.5 µ mand r min = 5.2 µ m, so that γ ( z ) = γ o / (9 . . πz/P )) (1Db, Fig. 3c). The observedintensity oscillations of the MMF field (MMGNLSE, Fig. 3b) are approximated better by8 o / (9 . . πz/P )) (space+time fit, Fig. 3b). This is because the MM soliton’sduration also oscillates: the soliton tends to lengthen when its spatial width increases andshorten when its spatial width decreases (Fig. 2a and inset, Supplementary Movie 1). It isthis spatiotemporal oscillation that generates the ODWs.The experimental peak locations are consistent with simulation and analytic theory. Fig-ure 3c shows the results of fitting the experimentally-measured and simulated peak locationswith the roots of Eq. 3 for various m . For the experimental (simulated) peaks, fitting yields β = -26 (-25) fs /mm, and β =143 (143) fs /mm. In both cases, the peaks are fit by thesimple theoretical approximation well, with the discrepancy attributed to the slightly differ-ent dispersion characteristics of the real and simulated fibers. The optimal values are nearthe simulation parameters, except that the optimal β is greater by 16%, which suggeststhat modal dispersion makes an effective contribution.Although the measured ODW positions are well-described, their amplitudes can varydue to several effects beyond both the simulations and analytic approach. First, due to thepresence of many modes in the fiber, phase-matched intramodal four-wave mixing (FWM)may occur involving the 1550-nm pump, its primary dispersive wave at 800-1000 nm, variousODWs, and Raman-shifted MM solitons. For the low-order ODWs in particular, we notethat there are several energy-conserving FWM processes. These may amplify red-shiftedODWs at the expense of specific blue-shifted ODWs. Particularly in the presence of manymodes, intramodal FWM can be a complex process[50], particularly when cascaded mixingis considered[51]. It is probably why, for certain initial spatial conditions, various low-orderblueshifted ODWs are attenuated or even invisible experimentally. We obtained the peaklocations in Figure 3c from experiments with multiple initial conditions, in order to accountfor low-order blue-shifted ODWs that were attenuated for any given intial condition. Sec-ond, because the spacing of the ODWs is quite close to the Raman bandwidth of fused silica,certain ODWs may experience Raman gain from one another and from the third-harmoniclight (THG: its efficiency may also benefit from intramodal phase-matching). Lastly, the dy-namic range and spectral resolution of the spectrometer limits the visibility of low-amplitudefeatures, and broadens narrow spectral features.Another remaining mystery is the relatively high amplitude of the ODWs observed inexperiments, compared to simulations. Intramodal FWM may play some role, as well aslarger oscillations and THG. Figure 3a shows the spectrum obtained by solving the 1D9onlinear envelope equation (NEE) with γ ( z ) = γ o / (15 . . πz/P )) , includingself-steepening and assuming averaging of the oscillating Raman integral[52]. Larger oscilla-tions produce relatively more intense dispersive waves, because the soliton is more stronglyperturbed. Larger oscillations occur when more modes are coherently locked together[22],and as the experiment contains much more than 5 modes, we choose a functional form of γ ( z ) to model this. In addition, THG provides an initial seed in the visible, leading to signif-icantly enhanced resonant emission at nearby wavelengths. Periodic backconversion occursfor frequencies near 3 ω o , so the largest enhancement is for slightly smaller frequencies.The ODWs generated by soliton resonances in MMF have relevance to applications.For example, by filtering the first redshifted or blueshifted ODW, one may generate pulsesin wavelength regions well outside the gain spectrum of available fiber dopants. Tuningmay be achieved by changing the pump wavelength or fiber pitch. In fact, because themodulation instability gain spectrum of a CW field at the pump wavelength overlaps withthe soliton sidebands[33], filtered ODWs could be parametrically amplified (using either thecirculating pump pulse or an injected CW field) provided some mechanism was introducedto compensate for the chromatic walk-off between the pump and ODW fields. ODWs spacedover a very broad wavelength range are generated coherently from the optical pump pulse.Given the remarkable range and coherence, spatiotemporal soliton oscillation may provide ameans of generating synchronized ultrashort pulses in different regions of the electromagneticspectrum. In an appropriate waveguide, an optically-derived coherent connection betweenmicrowave, deep ultraviolet and optical frequencies may be possible.In summary, we have shown that the spatiotemporal oscillation of nonlinear waves inGRIN multimode fiber causes the generation of their spatiotemporal dispersive waves. Thesedispersive waves can be described relatively well by simulations using the GMMNLSE,and insight can be gained by approximating the dynamics in a quasi-1D model with alongitudinally-varying nonlinearity. Future work, involving more advanced models and ex-perimental methods, can answer a few of the open mysteries about this process, includingverification of these hypotheses, the spatiotemporal dynamics leading to the multimodesupercontinuum, and the spatiotemporal structure of the dispersive waves. This work illus-trates valuable conceptual connections between 1D and the intermediate dimensional systemof the GRIN MMF. Practically, it provides a route to fiber-based ultrashort pulse sourceswith tunable wavelengths far outside the range of any current fiber-optic technique.10 EFERENCES [1] A. Hasegawa, Applied Physics Letters , 142 (1973).[2] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Physical Review Letters , 1095 (1980).[3] Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals [Hardcover] (Academic Press; 1 edition, 2003) p. 540.[4] A. Couairon and A. Mysyrowicz, Physics Reports , 47 (2007).[5] D. Faccio, M.A Porras, A. Dubietis, F. Bragheri, A. 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