Hof, K. D.

Klein, M. W.

Tritschler, T.

Wegener, M.

English

KITopen

April 1, 2005 / Vol. 30, No. 7 / OPTICS LETTERS 753First published in:Variation of the carrier-envelope phase offew-cycle laser pulses owing to the Gouy phase:a solid-state-based measurementT. Tritschler, K. D. Hof, M. W. Klein, and M. WegenerInstitut für Angewandte Physik, Universität Karlsruhe (TH), Wolfgang-Gaede-Strasse 1, 76131 Karlsruhe, GermanyReceived September 3, 2004The carrier-envelope phase of a laser pulse has recently become an important quantity in extreme nonlinearoptics. Because of the topological Gouy phase, it changes while the pulse propagates through the focus of alens. This variation is measured by a simple solid-state-based approach. The experimental results are ana-lyzed by comparison with simple analytical model calculations. © 2005 Optical Society of AmericaOCIS codes: 190.7110, 300.6470.When focusing a monochromatic laser by means of alens, the light field acquires a phase with respect to aplane wave while propagating along the optical axisszd. This Gouy phase1–3 has been known for morethan a century. Mathematically,4 it can be inter-preted as a topological phase in analogy with theBerry phase5 for electrons. An intuitive discussioncan be found in Ref. 6. Propagating from −` throughthe focus at z=0 to +`, the wave acquires a p phaseshift relative to a plane wave, which means that thecarrier wave propagates from −` to +` faster than aplane wave by half a cycle of light—the phase veloc-ity is superluminal, especially near the focus. On theother hand, the average group velocity of a laserpulse, i.e., the average velocity of the envelope, is lu-minal. Thus the carrier-envelope phase of a pulsechanges while propagating through the focus. Thiseffect becomes relevant in extreme nonlinear optics7in which the carrier-envelope phase generally playsan important role.The p phase shift caused by the Gouy phase wasdirectly time resolved in the terahertz regime a fewyears ago.8,9 More recently,10 the variation of thecarrier-envelope phase versus the propagation coor-dinate was measured in the optical regime with anamplified femtosecond laser system and a sophisti-cated analysis of photoelectron spectra obtained fromatoms ionized by the intense laser radiation withinthe focus. Nevertheless, it would still be interestinggenerally to have a simpler and specifically to have asolid-state-based approach suitable for measuringthe shift of the carrier-envelope phase originatingfrom the Gouy phase.In this Letter we present corresponding experi-ments with a phase-frequency-stabilized 5-fs laseroscillator and the recently introduced monolithiccarrier-envelope-frequency detection by means ofZnO crystals in the perturbative regime. The experi-mental results are compared with calculations of thecarrier-envelope phase based on the Gouy phase fordifferent focusing conditions.ZnO crystals indeed play a double role in the ex-periment (see Fig. 1). A fraction sù40 mWd of the to-tal average output power s140–180 mWd of our 5-fsprism-based laser oscillator (a copy of that described0146-9592/05/070753-3/$15.00 © EVA-STAR (Elektronisches Volltextarchhttp://digbib.ubka.uni-karlsruhe.de/volltin Ref. 11) operating at an fr=81 MHz repetition fre-quency is tightly focused onto a first 250-mm-thickZnO crystal with a linear polarization parallel to thecrystallographic cW axis12 (which has no center of in-version). The beat signal arising from the interfer-ence of self-phase modulation and the second har-monic, both generated within the ZnO crystal, isselected by means of optical filters (3-mm Schott BG39, 3-mm Schott GG 455, and Coherent 35-5289-000500-nm short-pass interference filter) and convertedinto a voltage signal by a 50-V-terminated photomul-tiplier tube (Hamamatsu R4332, Bialkali). Thiscarrier-envelope-frequency signal is fed into a feed-back loop that controls the angle of the end mirror inthe prism arm of the laser cavity by means of a piezo-electric actuator with an approximately 1-kHz band-width. This allows us to stabilize the carrier-envelopefrequency to ff=5 MHz, which is routinely monitoredwith a rf spectrum analyzer (Agilent PSA E4440A).The amplified and filtered photomultiplier signal isadditionally used as a reference for a rf lock-in am-plifier (Stanford Research Systems SR844).The remaining fraction of the laser output power isattenuated by means of homemade metal-coated100-mm-thin glass slides to an average power ofFig. 1. Scheme of the experimental setup. The output of a5-fs mode-locked laser oscillator is split into two balancedarms. In each arm the light is focused with a Cassegrainlens (L) onto a ZnO crystal, followed by a set of optical fil-ters (F), and a detector (D). One arm is used to stabilize thecarrier-envelope frequency of the laser and additionallyserves as the reference (REF) input for a lock-in amplifier.The other arm delivers the signal (SIG) input for thelock-in amplifier. The amplitude and phase of the output ofthe lock-in amplifier are recorded while scanning the ZnOcrystal of the signal arm through the focus.2005 Optical Society of Americaiv – Scientific Articles Repository) exte/1000010951 754 OPTICS LETTERS / Vol. 30, No. 7 / April 1, 2005P=6 mW, which is focused onto another250-mm-thick ZnO crystal by a Cassegrainian lens (amirror pair) with a focal length of 5.41 mm and a nu-merical aperture of 0.5 (Coherent 25-0522). The lightemitted in the forward direction is collected by an-other Cassegrainian lens (13.41-mm focal length, nu-merical aperture 0.5, Coherent 25-0555), spectrallyfiltered as for the reference signal, sent to another50-V-terminated photomultiplier tube (HamamatsuR4332, Bialkali), and fed into the signal input of therf lock-in amplifier. Importantly, the lock-in allowsfor measuring the amplitude and the phase of the rfsignal with respect to the rf reference.13 It is clearthat the rf lock-in phase in this arrangement directlyreflects the shift of the carrier-envelope phase of thelaser pulses—apart from an unknown but constantphase offset owing to the electronics and imperfectbalancing of the two arms.Figure 2(a) shows the beam width wsz ,v0d of afundamental Gaussian beam at laser carrier fre-quency v0 (with "v0=1.5 eV) within the Fresnelapproximation3 according tow2sz,vd = w02svdH1 + F zzRsvdG2J . s1dHere ws0,vd=w0svd is the beam waist in the focalplane at z=0, andzRsvd =w02svd2c0v s2dis the Rayleigh length at frequency v. The experi-mental Rayleigh length zRsv0d can be obtained in twoFig. 2. (a) Gaussian beam width wsz ,v0d versus propaga-tion coordinate z in units of the Rayleigh length zRsv0d=2.8 mm at laser carrier frequency v0. The open circles arethe result of a knife-edge measurement. The correspondingraw data are shown by the gray-scale plot in the back-ground. Here 50% of the maximum is set to black and 0%and 100% are set to white. (b) Lock-in phase (open circles),which is identical to the shift of carrier-envelope phasefszd. The solid curves correspond to the theoretical expec-tation for cases (i)-(iii). Case (ii) is identical to the Gouyphase. Amplitude A of the lock-in signal is shown by thegray area at the bottom.different and independent ways. Both are based on aleast-squares fit [solid curves in Fig. 2(a)] of wsz ,v0dto the experimental data [open circles in Fig. 2(a)]that are obtained from a knife-edge measurementalong the vertical direction. This fit delivers the twofit parameters w0sv0d=0.9 mm and zRsv0d=2.8 mm di-rectly. As a consistency check, zRsv0d can also be cal-culated with the above formula, inserting the value ofw0sv0d obtained from the fit. This delivers zRsv0d=3.0 mm, which is compatible with the above valuewithin the experimental errors. From a knife-edgemeasurement along the horizontal direction we ob-tain zRsv0d=2.4 mm. The curve labeled (ii) in Fig. 2(b)is the resulting Gouy phase wGsz ,v0d according towGsz,vd = arctanF zzRsvdG . s3dFor higher-order modes a different prefactor results.6The open circles in Fig. 2(b) correspond to the mea-sured lock-in phase for variation of position z of theZnO crystal while keeping the Cassegrainian lensesfixed. z.0 means that the front of the ZnO crystal isbehind the focus. The error bars reflect the statisticalerror of the average over ten individual scans. We de-pict only the phase in the central region of approxi-mately ±4 mm in which the lock-in amplitude signal(gray area) has appreciable strength. In this regionthe phase increases monotonically. This might seemsurprising at first considering that the ZnO crystal ismore than an order of magnitude thicker than theRayleigh length; hence a substantial averaging withrespect to the phase should arise. However, the co-herence length for phase matching of second-harmonic generation within the ZnO crystal can eas-ily be estimated to be of the order of 1 mm on thebasis of the known Sellmeier coefficients for ZnO.14Thus the effective sample thickness is indeed smallerthan the Rayleigh length. We briefly note that a morecomplex behavior of the phase versus z is observed atelevated powers incident onto the ZnO crystal of thesignal arm (also see Ref. 15). The data shown in Fig.2 are in the low-intensity limit.We now discuss the relation between the measuredcarrier-envelope phase [Fig. 2(b)] and the Gouy phaseshift. Generally, the carrier-envelope phase fszd ofthe laser electric field moving in the +z direction,quoted at a fixed position z on the optical axis,Estd = Ẽstdcosfv0t + fszdg, s4dis determined by the Gouy phase but it is not identi-cal to the Gouy phase. It results from the frequencydependence of the beam width wszf ,vd in front of the(ideal) lens at a focal length of z=zf@zRsv0d. This isan experimentally known fact16,17 that translatesinto a frequency dependence of the Rayleigh lengthzRsvd, which determines wGsz ,vd and hence fszd aswell as the electric field envelope Ẽstd. Closely follow-ing the discussion and the nomenclature of Ref. 7, wecan identify three special cases: (i) frequency-18independent beam width in front of the lens, (ii)April 1, 2005 / Vol. 30, No. 7 / OPTICS LETTERS 755frequency-independent Rayleigh length,19 and (iii)frequency-independent beam waist in the focus.20 Toobtain simple analytical results, we employ twoapproximations: First, we assume that the devia-tions of phase velocity vphase and group velocity vgroupfrom vacuum velocity of light c0 are small, allowingfor Taylor expansions. The validity of this approxima-tion follows from the Fresnel approximation madeabove. Second, we assume that the envelope of thepulse remains undistorted. In this limit the group ve-locity at the carrier frequency describes the velocityof the envelope. With this it is straightforward to ar-rive at the general form for the carrier-envelopephase:fszd =v0c02 E−`zfvphasesz8d − vgroupsz8dgdz8= wGsz,v0d − v0]wG]vsz,v0d. s5dUpon working out the derivative we obtainfszd = arctansZd + FZ1 + Z2, s6dwith the abbreviation Z=z /zRsv0d. Factor F is givenby F=−1 for case (i), F=0 for case (ii), and F= +1 forcase (iii). More generally, this expression for fszd re-mains valid even for arbitrary noninteger values ofF, in which case F is defined by the scalingzRsvd ~ vF. s7dThe scalings for the beam waist, w0svd~v+sF−1d/2, andthe beam width at the lens, wszf ,vd~v−sF+1d/2, follow.The three solid curves in Fig. 2(b) illustrate cases(i)–(iii). Only for case (ii), where vgroupszd=c0, is thecarrier-envelope phase identical to the Gouy phaseshift, i.e., fszd=wGsz ,v0d. For case (i), fszd has a hori-zontal slope in the focus because vgroupsz=0d=vphasesz=0d.c0. For case (iii) the slope of fszd in thefocus is steeper than in case (ii) as vgroupsz=0d,c0while vphasesz=0d.c0.Experimentally, it is usually not easy to specify thefrequency dependence of the beam width wszf ,vd infront of the lens because it results from rather com-plex processes within the mode-locked laser oscilla-tor. It would be surprising if this frequency depen-dence turned out to be negligible [case (i)]. Ourexperiment lies between cases (ii) and (iii).In conclusion, we have realized a simple solid-state-based technique suitable for measuring thevariation of the carrier-envelope phase of few-cyclelaser pulses in the focus of a lens arising from the to-pological Gouy phase. The approach is based on thewell-known f-to-2f self-referencing. Importantly, wehave generated the corresponding beat signal in a re-gion thinner than the Rayleigh length, such that thelocal carrier-envelope phase can be measured. Fur-thermore, unlike other approaches based on multi-photon ionization of atoms,10 our approach workswith pulses directly from a usual 81-MHz mode-locked laser oscillator. We have also shown by simpleanalytical formulas that the evolution of the carrier-envelope phase critically depends on the frequencydependence of the laser beam profile. This frequencydependence is expected to be affected not only by thelaser system but also by its actual alignment. Our ex-perimental approach allows for a simple routinecharacterization of the resulting local carrier-envelope phase. In the emerging field of extreme non-linear optics, such knowledge concerning the carrier-envelope phase is crucial.The work of M. Wegener has been supported byproject DFG-We 1497/9-1. We thank F. X. Kärtnerand U. Morgner for providing the double-chirped mir-rors. T. Tritschler’s e-mail address is thorsten.tritschler@physik.uni-karlsruhe.de.References1. C. R. Gouy, Acad. Sci. Paris C. R. 110, 1251 (1890).2. C. R. Gouy, Ann. Chim. Phys. 24, 145 (1891).3. A. E. Siegmann, Lasers (University Science, MillValley, Calif., 1986); note that the sign convention usedfor the Gouy phase in this textbook is different fromthe one used in this Letter.4. D. Subbarao, Opt. Lett. 20, 2162 (1995).5. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).6. S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).7. M. Wegener, Extreme Nonlinear Optics (Springer-Verlag, Berlin, 2004).8. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H.G. Winful, Phys. Rev. Lett. 83, 3410 (1999).9. E. Budiarto, N.-W. Pu, S. Jeong, and J. Bokor, Opt.Lett. 23, 213 (1998).10. F. Lindner, G. G. Paulus, H. Walther, A. Baltuška, E.Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev.Lett. 92, 113001 (2004).11. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A.Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G.Angelow, and T. Tschudi, Opt. Lett. 24, 411 (1999).12. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner,and F. X. Kärtner, Opt. Lett. 27, 2127 (2002).13. D. J. Jones, T. M. Fortier, and S. T. Cundiff, J. Opt.Soc. Am. B 21, 1098 (2004).14. X. W. Sun and H. S. Kwok, J. Appl. Phys. 86, 408(1999).15. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner,and F. X. Kärtner, Phys. Rev. Lett. 89, 127401 (2002).16. S. T. Cundiff, W. H. Knox, E. P. Ippen, and H. A. Haus,Opt. Lett. 21, 662 (1996).17. T. M. Fortier, D. J. Jones, and S. T. Cundiff, Opt. Lett.28, 2198 (2003).18. Z. L. Horváth, J. Vinkó, Zs. Bor, and D. von der Linde,Appl. Phys. B: Lasers Opt. 63, 481 (1996).19. S. Feng, H. G. Winful, and R. W. Hellwarth, Phys. Rev.E 59, 4630 (1999).20. A. E. Kaplan, J. Opt. Soc. Am. B 15, 951 (1998).

Variation of the carrier-envelope phase of few-cycle laser pulses owing to the Gouy phase: a solid-state-based measurement

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Variation of the carrier-envelope phase of few-cycle laser pulses owing to the Gouy phase: a solid-state-based measurement

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