Nnlinear Analysis: Mdelling and Cntrl, Vilnius, IMI,, N 5 Lithuanian Assciatin f Nnlinear Analysts, Slitn-Effect Optical Pulse Cmpressin in Bulk Media with χ (3) Nnlinearity Received: 9.7. Accepted: 11.1. G. Tamšauskas, A. Dubietis and G. Valiulis Department f Quantum Electrnics, Vilnius University Saulėteki al. 9, LT-4 Vilnius Lithuania [gintaras.tamsauskas. gintaras.valiulis]@ff.vu.lt Abstract Self-cmpressin f visible ptical pulse in bulk χ (3) medium has been demnstrated taking the advantage f negative grup-velcity dispersin f tilted pulses. Keywrds: tempral slitns, tilted pulses, self-phase-mdulatin, pulse cmpressin 1 Intrductin Optical pulse cmpressin is well-established technique fr pwerful ultrashrt pulse generatin. The technique is based n the pulse chirping in the nnlinear media and the chirp cmpensatin utilizing dispersin btained by gratings r prisms. Using bulk (free f aperture limitatins) materials, pulse chirping under psitive grup-velcity dispersin (GVD) and cmpressin was demnstrated mre than a decade ag [1]. An alternate apprach is t cmbine self-phase-mdulatin (SPM) and negative GVD utilizing the slitn cmpressin effect. Experimental evidence f slitn-effect pulse cmpressin dates back t early 8`s and was preferably studied in ptical fibers with negative GVD [,3]. A cmbinatin f SPM and negative GVD in bulk materials has been nly cnsidered theretically in the sense f generatin f light bullets [4] and directed t the case f wavelengths f arund 1.5 µm that reach an anmalus dispersin regin fr fused silica and glass. Hwever, the methd is nt widely applied fr the pulse cmpressin because f limited wavelength range (~1.5µm) and absence f well-develped laser surces emitting at this wavelength range. 99
Recently, tempral slitn frmatin was achieved in visible [5] and near IR [6] by use f tilted-frnt pulses and expliting χ () and χ (3) nnlinearities f the ptical crystals. Mre extended theretical study [7] pinted that the negative GVD cnditin is readily achievable fr a wide range f wavelengths and ptical materials by apprpriate pulse-frnt tilting. Mrever, it has been shwn that pure χ (3) nnlinearity als cntributes fr the tempral slitn frmatin alng with χ () nnlinearity being the driving ne. Numerical Mdel and Cmputer Simulatin The pulse prpagatin in a dispersive medium with instant χ (3) nnlinearity is gverned by the nnlinear Shredinger equatin that in nedimensinal case can be expressed as 1 k A A A i A + g = in z u t t where u is the pulse grup velcity, A, (1) g = ( d k / dω ) is the GVD ω = ω cefficient, ω is the carrier frequency, k =ω /c is the wave number and (3) n = ( π / n ) χ is the nnlinear refractin index satisfying the equatin n=n +n I with I being the applied intensity. The light pulse is called tilted when it pssesses a tempral delay acrss the transverse crdinate [8]. This means that the pulse frnt (the surface f cnstant intensity at a fixed time) is nt parallel t the phase frnt (the surface f cnstant phase at a fixed time) and hence is nt perpendicular t the directin f prpagatin. Usually the pulse frnt tilt is btained by use f ptical elements that intrduce an angular dispersin, i.e. diffractin gratings r prisms. The GVD cefficient f the medium is then g T 1 tanα = g, k u where subscript T refers t the tilted pulse, and α is the pulse-frnt tilt angle. In Ref. 7 it was shwn that in the large-beam apprximatin the prpagatin f a tilted pulse culd be described by the Eq.1 with mdified dispersin f the medium, accrding t the effective values given in Eq.. Obviusly, the pulse-frnt tilting results in a material dispersin cmpensatin r even in the anmalus dispersin. Fig. 1 illustrates the GVD cefficient f BK7 glass versus the tilt angle α fr 57-nm, τ=165 fs pulses. The tilt angles starting frm ~13 deg result in anmalus dispersin f BK7 glass (negative GVD cefficient). The nnlinear refractive index fr BK7 glass () 1
n =.8 1 - m /W was taken frm Ref. 9. If the pulse intensity is high enugh fr SPM t ccur, these GVD> 3 g, 1 fs/m T - -4-6 -8 GVD< -1 1 3 Pulse-frnt tilt angle, deg Fig. 1.Grup velcity dispersin cefficient f BK7 glass as functin f the pulse-frnt tilt angle α, calculated fr 165-fs pulses. cnditins are suppsed t cause a slitn-effect pulse cmpressin. The main differences as cmpared t the ptical fiber case are the fllwing: 1) the pulse prpagates freely withut aperture limitatin; ) negative grup velcity dispersin is predetermined by the pulse tilt rather than by the material dispersin; 3) due t angular dispersin the tilted pulses will nt peridically recnstruct the tempral shape while prpagating in the media. Pulse duratin, fs 3 5 15 1 5 α=8, z=3mm α=35, z=3mm α=35, z=16mm α=8, z=16mm 1 3 4 5 6 Incident pulse intensity, GW/cm Fig.. Pulse duratin dependence n the incident intensity fr different pulsefrnt tilt angles and media length z. 11
The simulatins were perfrmed by numerically slving Eq.1 in the plane wave apprximatin. Results f numerical simulatins are presented in Fig.. Fr lw Intensity, GW/cm 1 8 6 4 z= mm z=11 mm z= mm -. -.1..1. Time, ps Fig. 3. Transfrmatin f the pulse prfile during the prpagatin thugh BK7 glass. Incident pulse frnt is tilted by 8 deg and I=3 GW/cm. intensity pulses the negative GVD dminates ver the SPM, and the result is pulse dispersive bradening. By increasing the input intensity the SPM drives the pulse t cmpress and at certain input intensity the SPM and negative GVD cmpensate fr each ther resulting in chirp-free shrt (<5 fs) pulse. Further intensity increasing leads t frmatin f higher rder slitns. The typical transfrmatin f pulse tempral prfile is shwn in Fig.3. 3 Experiment The 165-fs pulse at 57 nm were delivered by fiberless CPA Nd:glass laser (TWINKLE, Light Cnversin Ltd.), equipped by the nnlinear SH pulse cmpressr [1]. The utput beam has diameter f 6 mm at FWHM and ttal energy f 3 mj. The input pulse was tilted by 6 mm -1 diffractin grating perating at secnd diffractin rder clse t Littrw cnditin. The grating G1 was imaged by a telescpe T1 nt the input face f 16-mm-lng BK7 slab with beam size reductin factr f 1.66. The grve spacing f the grating and telescpe magnificatin were chsen t achieve a tilt angle f 35 deg inside the glass slab. The identical set f ptics was used fr canceling the pulse-frnt tilt. The spacing between gratings and imaging ptics was aligned fr zer dispersin; it was justified by cmparing input and utput pulsewidths. In the presence f BK7 slab, the rear part f the setup (i.e. telescpe T and grating G which were used t restre the untilted pulse) was realigned t image the 1
exit face f the BK7 slab nt the secnd grating. In that way the net dispersin f whle arrangement was set by the dispersin prduced thrugh the pulsefrnt tilting as derived frm Eq.. The accessible range f intensities with all factrs accunted (grating diffractin efficiency, beam-size reductin by imaging telescpe, etc.) was up t 5 GW/cm. G1 Pl λ/ input f1 f f f1 G T1 T BK7 utput f4 Fig. 4. Experimental setup. G1, G diffractin gratings; f1=f4=5 mm, f=f3=3 mm, λ/ half-wave plate, P plarizer, fr the incident intensity adjustment. With increasing the incident pulse intensity, spectrum bradening and pulse self-cmpressin were bserved. At the pump intensity f ~35 GW/cm the cmpressin yielded a pulse with autcrrelatin width f 135 fs, see Fig. 5. Assuming the autcrrelatin/pulsewidth factr f 1.4, which was btained by numerical mdelling, the duratin f the self-cmpressed pulse was evaluated t be 95 fs. The Furier transfrm f the measured spectrum revealed almst the same value, pinting t absence f the residual chirp. Qualitatively the pulsewidth dependence n the incident intensity fllwed the theretical predictins; hwever, the bserved pulses had smewhat lnger duratin. The duratin f the cmpressed pulse versus the input intensity is depicted in Fig. 6. The quantitative discrepancies between the numerics and the experiment may be explained as fllws. At higher intensities (>4 GW/cm ) spatially dispersed white light cntinuum was bserved. The theretical mdel als des nt take int accunt the effect f stimulated Raman scattering whse impact may be mre cmplex than just lss f intensity [11]. Anther nnlinear prcess that was nt taken int accunt was the tw-phtn absrptin. Fr high intensities (abve 5 GW/cm ) nnlinear lsses cnsumed up t % f the incident pulse energy. These three afrementined prcesses exhibit cmplex nnlinear lss mechanism and affect mstly the tp f the pulse, which leads t the increase f pulse duratin. f4 f3 f3 length adjustment 13
1..8 u a.6 y nsit.4 e nt I. τ crr =3 fs (a) (b). 1..8 u a.6 y n sit.4 e nt I. -1-5 5 1 τ crr =135 fs (c) 5 5 54 56 58 53 53 (d). -1-5 5 1 5 5 54 56 58 53 53 Delay, fs Wavelength, nm Fig. 5. Autcrrelatin traces and spectra f incident (a), (b) and selfcmpressed (c), (d) pulses. The incident pulse spectrum (b) with tw peaks is the characteristic ne fr nnlinear SH pulse cmpressin [9]. Autcrrelatin width, fs 35 3 5 15 1 Incident intensity, GW/cm Fig. 6. Autcrrelatin width dependence n the incident intensity. Anther shrtcming f the theretical mdel is that it des nt take int accunt the spatial distributin f the beam (plane-wave mdel was used). At finite beam diameter and near Gaussian intensity distributin high intensity 14 experiment simulatin 5 1 3 4 5
beam must experience sme nset f self-fcusing. This means that the directin at which a particular spectral cmpnent prpagates depends nt slely n its wavelength but als n the curvature f spatial intensity prfile at the spt it riginates frm. In cnclusin, the slitn-effect pulse cmpressin in bulk χ (3) medium has been experimentally demnstrated. Anmalus dispersin fr 57-nm pulses in BK7 glass was achieved by an apprpriate pulse-frnt tilting. Cmpressin f 165-fs Nd:glass secnd-harmnic pulse dwn t 95 fs was bserved. References 1. C. Rlland and P. B. Crkum, Cmpressin f high-pwer ptical pulses, J. Opt. Sc. Am. B 5, 641-647 (1988).. L. F. Mllenauer, R. H. Stlen, and J. P. Grdn, Experimental bservatin f picsecnd pulse narrwing and slitns in ptical fibers, Phys. Rev. Lett. 45, 195-198 (198). 3. L. F. Mllenauer, R. H. Stlen, J. P. Grdn, and W. J. Tmlinsn, Extreme picsecnd pulse narrwing by means f slitn effect in singlemde ptical fibers, Opt. Lett. 8, 89-91 (1983). 4. Y. Silberberg, Cllapse f ptical pulses, Opt. Lett. 15, 18-184 (199). 5. P. Di Trapani, D. Cairni, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, Observatin f tempral slitns in secnd-harmnic generatin with tilted pulses, Phys. Rev. Lett. 81, 57-573 (1998). 6. X. Liu, L. J. Qian, and F. W. Wise, Generatin f ptical spatitempral slitns, Phys. Rev. Lett. 8, 4631-4634 (1999). 7. G. Valiulis, A. Dubietis, R. Danielius, D. Cairni, A. Viscnti, and P. Di Trapani, Tempral slitns in χ () materials with tilted pulses, J. Opt. Sc. Am. B 16, 7-731 (1999). 8. O. E. Martinez, Pulse distrtins in tilted pulse schemes fr ultrashrt pulses, Opt. Cmmun. 59, 9-3 (1986). 9. D. N. Nikgsyan, Prperties f ptical and laser-related materials, Wiley, Chichester, p.37 (1997). 1. A. Dubietis, G. Valiulis, R. Danielius, and A. Piskarskas, Nnlinear pulse cmpressin by ptical frequency mixing in crystals with secnd-rder nnlinearity, Pure Appl. Opt.. 7, 71-79 (1998). 11. K. Ch. Chan and H. F. Liu, Effect f third-rder dispersin n slitn effect pulse cmpressin, Opt. Lett.19, 49-51 (1994). 15