a r X i v :p h y s i c s /0303083v 1 [p h y s i c s .o p t i c s ] 20 M a r 2003
Spontaneously generated X-shaped light bullets
P.Di Trapani,1G.Valiulis,2A.Piskarskas,2,O.Jedrkiewicz,1J.Trull 1,C.Conti,3S.Trillo,3,4
1
INFM and Department of Chemical,Physical and Mathematical Sciences,
University of Insubria,Via Valleggio 11,22100Como,Italy 2
Department of Quantum Electronics,Vilnius University,Sauletekio al.9,bldg.3,LT-2040Vilnius,Lithuania 3
Istituto Nazionale di Fisica della Materia (INFM)-RM3,Via della Vasca Navale 84,00146Roma,Italy and
4
Department of Engineering,University of Ferrara,Via Saragat 1,44100Ferrara,Italy
(Dated:February 2,2008)
We obrve the formation of an inten optical wavepacket fully localized in all both longitudinally (in time)and in the transver plane,with an extension of a few tens of fc and microns,respectively.Our measurements show that the lf-trapped wave is a X-shaped light bullet spontaneously generated from a standard lar wavepacket via the nonlinear material respon (i.e.,cond-harmonic generation),which extend the soliton concept to a new realm,where the main hump coexists with conical tails which reflect the symmetry of linear dispersion relationship.
PACS numbers:03.50.De,42.65.Tg,05.45.Yv,42.65.Jx
Defeating the natural spreading of a wavepacket (WP)is a universal and challenging task in any physical context involving wave propagation.Ideal particle-like behavior of WPs is demanded in applications,such as microscopy,tomography,lar-induced particle acceleration,ultra-sound medical diagnostics,Bo-Einstein condensation,volume optical-data storage,optical interconnects,and tho encompassing long-distance or high-resolution sig-nal transmission.The quest for light WPs that are both invariant (upon propagation)and sufficently localized in all dimensions (,both transversally and longitu-dinally or in time)against spreading ”forces”exerted by diffraction and material group-velocity dispersion (GVD,k ′′=d 2k/dω2|ω0)has motivated long-standing stud-ies,which have followed different strategies in the linear [1,2,3,4,5]and nonlinear [6,7]regime,respectively.In the linear ca,to counteract material (intrinsic)GVD,one can exploit the angular dispersion (i.e.,de-pendence of propagation angle on frequency)that stems from a proper WP shape.The prototype of such WPs is the X-wave [2],a non-monochromatic,yet non-dispersive,superposition of non-diffracting cylindrically symmetric Besl J 0(so-called conical or Durnin [1])beams,ex-perimentally tested in acoustics [3],optics [4]and mi-crowave antennae [5].Importantly,in the relevant ca of WPs with relatively narrow spectral content both tempo-rally (around carrier frequency ω0)and spatially (around propagation direction z ,i.e.paraxial WPs),X-waves require normally dispersive media (k ′′>0).In this ca,a WP with disturbance E (r,t,z )exp(ik 0
z −iω0T )(k 0≡k (ω0),r 2≡x 2+y 2),has a slowly-varying envelope E =E (r,t,z )obeying the standard wave equation
ˆL世故是什么意思
(ω0)E =0;ˆL (ω0)≡i∂z +12
∂2
tt .
(1)
Laplacian,where ∇2⊥=∂2rr +r
−1
∂r is the transver Laplacian,and we limit our attention to luminal WPs
traveling at light group-velocity 1/k ′=dk/dω|−1
ω0by
FIG.1:Features of normally dispersive media (k ′′>0):(a)
spatio-temporal dispersion relation associated with paraxial wave equation (1);Different reprentations [(b)ctions of color isointensity surfaces |E (x,y,t )|2=const.;(c)intensity vs.r,t ]of a non-diffractive,non-dispersive linear X-wave (∆=10fs,k ′′=0.02ps 2/m,k 0=107m −1).
走自己的路让别人说去吧introducing the retarded time t =T −k ′z in the
WP barycentre frame.Propagation-invariant waves E (r,t,z )=E (r,t,z =0)exp(iβz )can be achieved when-ever their input spatio-temporal spectra E (K,Ω,z =0)lie along the characteristics of the dispersion relationship k ′′Ω2/2−K 2/(2k 0)=β,which follows from Eq.(1)in Fourier space (K,Ω)(K is the transver wavevector re-lated to cone angle with z -axis θ≃sin θ=K/k 0,and Ω=ω−ω0).In the normal GVD regime (k ′′>0)the curves,displayed in Fig.1(a),reflect the hyper-bolic nature of the wave equation (1)and show the com-mon asymptotic spectral X-shape associated with the lines K =±2k 0k ′′Ω(β=0).When the spectral com-ponents lying along such X are superimpod coherently
(in pha),thefield in the physical space(r,t)also re-tains a propagation invariant X-shape,as by the exact solution E=Re{[(∆−it)2+k0k′′r2]−1/2} of Eq.(1)shown in Fig.1(b-c).Here∆reprents dura-tion of the X-wave central hump.Main features of the waves are the conical(clepsydra)3D str
ucture and the slow spatial decay(1/r characteristic of J0components [1]),displayed by Fig.1(b)and(c),respectively. Converly,in the nonlinear(high intensity)regime, non-spreading WPs exploit the idea that,in lf-focusing media,the nonlinear wavefront curvature can simulta-neously balance the curvature due to diffraction and to GVD,combining features of spatial[9]and tempo-ral[10]solitons to form a bell-shaped3D-localized WP E(r,t),so-called light-bullet[6].In sharp contrast with X-waves,such compensation strictly requires anomalous GVD(k′′<0)[6],thus implying that along the WP tails, where Eq.(1)still holds true,space r and time t play the same role giving ri to strong WP localization[13].Sta-ble trapping,however,has been obrved only in tting of reduced dimensionality(2D)[7,8],including also other spin[11]or atomic waves[12](3D kinetic energy and atom-atom attractive interactions act exactly as diffraction-GVD and lf-focusing,respectively)where similar trapping mechanisms hold true.
In this work,we outclass the two approaches by demon-strating that space-time localization in the normal GVD regime becomes accessible in the nonlinear regime.Trap-ping is accomplished by mutual balance of intrinsic, shape-induced,and nonlinear contributions in a new type of WP,namely a nonlinear X-wave,which permits to get over two limitations at once.First,in contrast with lin-ear X-waves,who obrvation requires non-trivial input beam shaping[3,4],our experiment reveals a rem
ark-able”mode-locking”process that,starting from a con-ventional(gaussian)lar WP,spontaneously performs the reshaping into a localized X-shaped WP.Second,we believe this to be thefirst genuine nonlinear trapping in full-dimensional3D physical space,since to date mate-rial and/or instability limitations[6,7]have rendered the obrvation of light3D bullets elusive.
Figure2describes the strong localization features ob-rved after propagation in a22mm long sample of lithium triborate(LBO)χ(2)crystal.At the input we launch a lar WP at fundamental frequency(FF)ω0=2πc/λ0,λ0=1060nm,with gaussian profile in both t(with FWHM duration in the100−200fs range) and r(45µm FWHM at waist,located few mm before the crystal so that the input beam is slightly diverging).The LBO crystal is tuned for generation of optical cond-harmonic(SH)in the regime of relatively large positive pha mismatch∆k=2k(ω0)−k(2ω0)=30cm−1or effective lf-focusing for the FF beam[15].When the input energy exceeds about0.25µJ,mutually trapped localized WPs at FF and SH are obrved(e Fig.2). Time-integrated measurements of spatial profiles(Fig.2, top)indicate that diffraction is fully defeated to yield a spatial soliton-like beam[9],while temporal autocor-
propagates(locked with the SH)with immutable shape and nearly constant energy,duration,and size.我的大学生活
By changing the parameters in Eqs.(2)we can con-clude that:(i)nonlinearity is the key element that drives the reshaping and holds the WP together.In fact,by switching it offafter the ,χ=0for z>20 mm),the WP exhibits strong diffraction and extremely
fast FF-SH walk-off.The reshaped WP can by no means be considered a linear X-wave;(ii)in our crystal,GVM is the dominant dispersion term affecting the asymptotic duration and width of the localized WP(both decrea for smaller GVM).However,(symmetric)X-shaped WPs are formed also in the ideal ca of vanishing GVM, provided that GVD is normal;(iii)the WP reshaping strongly affects the pha modulation process,which re-veals the contribution of an effective anomalous GVD; (iv)by including additional cubic nonlinearities[15],the phenomenon remains qualitatively unchanged.
A more rigorous ground for explaining the dramatic reshaping shown in Fig.3is offered by the investiga-tion of two different aspects,both implicitly accounted for by Eqs.(2):(i)the stability of continuous plane-waves;(ii)the existence of nonlinear X-shaped eigen-solutions.Though somewhat idealized,the results of this analysis allow gathering the two basic features of the spatio-temporal dyn
amics,namely the transient re-shaping of the input gaussian-like WP followed by the quasi-stationary regime.
长大成人2
The linear stability analysis of z-invariant solutions of Eqs.(2)with ideally vanishing spatio-temporal spectral ,continuous plane waves)always reveal the prence of exponentially growing weak(up to noi level) perturbations with definite frequency K,Ω.However, the instability features reflect the symmetry of the linear wave equation,thus being qualitatively different in the normal and anomalous GVD regime,respectively.While anomalous GVD leads to narrow bandwidth features as in conventional spatial or temporal modulational insta-bility[15],the hyperbolic structure of the diffraction-dispersion operator in the normal GVD regime of our experiment leads to exponential amplification of conical wave perturbations(Besl J0beams)with frequencies basically approaching the asymptotes of Fig.1(a)[16]. When growing spontaneously from noi,the amplified components in the virtual infinite bandwidth(in real-ity limited by non-paraxiality and higher-order dispersion not accounted for in Eqs.(1-2))leads to colored conical emission.Our calculations show that the phenomenon persists under dynamical conditions(uneded SH gen-eration)and when pumped by short-pul narrow-beam inputs,as in our experiment.In the latter ca,the am-plification of proper frequency components of the WP prerves the mutual pha coherence,thus acting as a trigger whi
ch drives the transformation of the WP into the X-wave shown in Fig.3.In other words,it is the conical instability that probes the symmetry of the un-derlying linear system in amplifying tho components which allow both diffraction and dispersion of the whole
100
X [µm]
100
200 Time [fs]
100
100
200
Time [fs]
100
搓捻X [µm]
100
200
Time [fs]
X [µm]
0 fs70 fs
(b)(c)
(a)
FIG.4:(a)Output spatio-temporal intensity profile,as mea-sured in air,5mm from the crystal output face.Ints:trans-ver intensity pattern in(x,y)plane measured at peak(t=0 fs)and with t=70fs delay.(b)As in(a),numerical result from Eqs.(2).(c)As in(b)calculated right on crystal output.
with data obtained numerically from Eqs.(2).However, it is only the tomography of the output
map-ping the WP intensity in space and time,that can give direct unequivocal evidence for the formation of an X-wave.To this end we have developed a new technique bad on an ultrafast nonlinear gating,or a scanning cross-correlation technique,realized by frequency mixing the WP under investigation with a20fs,high contrast, steep front,probe pul in the visible,which is uniform over a large(few mm2)area(the details of the t up will be prented elwhere).Thanks to the high quality of the probe and the u of a very thin(20µm)BBO mixing crystal,the apparatus has high temporal resolu-tion.The intensity map of the output WP is reported in Fig.4(a).The ints also show the measured beam profile in the transver plane at time t=0(peak)and t=70fs(far from peak),respectively.The measured profile clearly shows the features of an X-wave with a conical structure,which emanates from a strongly local-ized central spatio-temporal hump.Unlike conventional pul splitting[14],here splitting occurs only sufficiently off-axis(x∼100µm)in the WP low-intensity portion. For comparison we also report in Fig.4(b)the profile calculated from Eqs.(2)under the same conditions,the agreement being excellent.The fringe-like structure that appears for large delays in both Fig.4(a)and(b)is due to5mm of free-space propagation in air outside the crys-tal.Although the calculated profile on the output face of the crystal indeed shows that such fringes disappear [e Fig.4(c)],measurement with perfect imaging on the output LBO face reveals saturation of the mixing process due to a too inten peak.
In summary,we have reported thefirst evidence that the natural3D(temporal and spatial)spreading of a fo-cud ultrashort wavepacket can be balanced in trans-parent materials at high intensity.The underlying mech-anism is the spontaneous formation of a X-wave charac-terized by an ,nonlinear)central hump lf-trapped through mutual balance with(esntially linear) dispersive contributions associated with coexisting slowly decaying conical tails.While our experiment is carried out by exploiting lf-focusing nonlinearities arising from quadratic nonlinearity,we envisage the general role that lf-trapping mediated by nonlinear X-waves can have for a wide class of materials and applications encompassing centrosymmetric optical(Kerr)media[6],Bo-Einstein condensation[12],and acoustics[3].
We acknowledge support from MIUR(PRIN and FIRB projects),Unesco UVO-ROSTE(contract875.586.2), Lithuanian Science and Studies Foundation(grant T-491),Secretaria de Estado y Universidades in Spain,and Fondazione Tronchetti Provera in Italy.We are grateful to the technical assistance of Light Conversion Ltd.
[1]J.Durnin,J.J.Miceli,and J.H.Eberly,Phys.Rev.Lett.
58,1499(1987).
[2]E.Recami,Physica A252,586(1998);Salo,J.Fager-
holm,A.T.Friberg,and M.M.Salomaa,Phys.Rev.Lett.
83,1171(1999);J.Salo,J.Fagerholm,A.T.Friberg,and M.M.Salomaa,Phys.Rev.E62,4261(2000).
[3]J.Lu and J.F.Greenleaf,IEEE Trans.Ultrason.Ferrelec.
<39,441(1992).
[4]P.Saari and K.Reivelt,Phys.Rev.Lett.79,4135(1997);
H.S¨o najalg,M.Rtp,and P.Saari,Opt.Lett.22,310
(1997).
[5]D.Mugnai, A.Ranfagni,and R.Ruggeri,Phys.Rev.
Lett.84,4830(2000).
livingroom[6]Y.Silberberg,Opt.Lett.15,1282(1990);F.Wi and P.
Di Trapani,Opt.Phot.News13,28(2002)and references therein.
[7]X.Liu,l.J.Quian,and F.W.Wi,Phys.Rev.Lett.82,
4631(1999).X.Liu,K.Beckwitt,and F.W.Wi,Phys.
Rev.Lett.85,1871(2000).
[8]H.S.Einberg,R.Morandotti,Y.Silberberg,S.Bar-Ad,
D.Ross,and J.S.Aitchison,Phys.Rev.Lett.87,043902
(2001).
[9]S.Trillo and W.E.Torruellas,eds.,Spatial Solitons
(Springer,Berlin,2001).
[10]L.F.Mollenauer,R.H.Stolen,J.P.Gordon,Phys.Rev.
Lett.45,1095(1980).
[11]M.Bauer,O.Buttner,S.O.Demokritov,B.Hillebrands,
V.Grimalsky,Y.Rapoport,and A.N.Slavin,Phys.Rev.
Lett.81,3769(1998).
[12]L.Khaykovich,F.Schreck,G.Ferrari,T.Bourdel,J.Cu-
bizolles,L.D.Carr,Y.Castin,and C.Salomon,Science 296,1290(2002);K.E.Strecker,G.B.Partridge, A.G.
Truscott,and R.G.Hulet,Nature417,150(2002). [13]Mathematically the strong localization is associated with
existing solutions E(r,t)exp(iβz)of Eq.(1)with k′′<0,β>0,which decay exponentially in r,t→∞.
[14]J.K.Ranka,R.W.Schirmer,and A.L.Gaeta Phys.Rev.
暮光之城3剧情介绍Lett.77,3783(1996).
[15]A.V.Buryak,P.Di Trapani,D.Skryabin,and S.Trillo,
北京中考时间2021年具体时间Phys.Rep.370,63(2002).
px是什么意思[16]S.Trillo, C.Conti,P.Di Trapani,O.Jedrkiewicz,J.
numlockTrull,G.Valiulis,G.Bellanca,Opt.Lett.27,1451 (2002).
[17]C.Conti,S.Trillo,P.Di Trapani,G.Valiulis, A.Piskarskas,O.Jedrkiewicz,J.Trull,Los Alamos National Lab.e-print physics/0204066(2002).
