Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, England
Received July 3, 2001; revised manuscript received January 28, 2002
The ac-Stark shift of the A 2S1 n8 5 2 ← X 2Pr n9 5 0 two-photon Bohr resonance of nitric oxide at 409.8 nmis utilized to autocorrelate intense, ultrashort optical pulses at 400 nm. When they are temporally and spa-tially overlapped, two identical pulses shift the absorption into transient resonance with the applied two-photon energy. Interferometric autocorrelation traces are obtained by detection of A 2S1 n8 5 2 → X 2Pr
1. INTRODUCTIONThere are today a wide range of techniques available forcharacterizing ultrashort laser pulses. Most notably,spectral phase interferometry for direct electric fieldreconstruction1 (SPIDER), the several variants offrequency-resolved optical gating2 (FROG), and its de-rivative procedures temporal analysis by dispersing apair of light E-fields3 (TADPOLE) and grating-eliminatedno-nonsense observation of ultrafast incident laser lightE-fields4,5 (GRENOUILLE) offer a full determination ofthe duration and spectral phase of an ultrafast laser pulseof arbitrary shape.6 Predating these relatively recent ad-vances is the venerable technique of measuring the auto-correlation integral,7 whereby the duration of a lightpulse is determined, for an assumed envelope shape, fromthe instantaneous response of a nonlinear medium to anultrashort pulse overlapped in space and time with an ex-act replica of itself. Examples of autocorrelation proce-dures are too numerous to discuss at length here:second-harmonic generation (SHG) in phase-matchedcrystals,8 near-surface SHG,9,10 and measurement of two-photon currents in semiconductor photodiodes,11,12 toname but a few, are some of the nonlinear effects thathave been adopted for determining autocorrelationwidths of ultrafast laser pulses at visible wavelengths.For extension of the autocorrelation principle to the ultra-violet and extreme-ultraviolet ranges, two-photon phe-nomena such as ionization of rare gases,13,14 absorption inthin diamond crystals,15 excitation of self-trappedexcitons,16 photoacoustic spectroscopy,17 and conductivitymeasurements18 have been adopted as an autocorrelation
diagnostic of femtosecond laser pulses. Third-order phe-nomena such as the optical Kerr effect19–21 and cross-correlation techniques based on sum-frequency22 anddifference-frequency23 generation also find application fortemporal characterization of ultraviolet ultrafast pulses.
Although it is sometimes thought that it is impossibleto determine the variation of the spectral phase across alaser pulse by using autocorrelation procedures, such in-formation can be extracted by temporal decorrelation ofan autocorrelation trace combined with a measurement ofthe pulse spectrum.24 Irrespective of the origin of thenonlinear phenomenon that is adopted to characterize anultrashort pulse, in an autocorrelation measurement thetime delay tD between a signal pulse and a spatially over-lapped replica is scanned to map out an nth-order auto-correlation signal, which we write in the form25
An~tD! 5 E2`
`
u@Es~t ! 1 Er~t 2 tD!#2undt, (1)
where Es(t) 5 Es0(t)exp@ivs t 1 ifs(t)# is the signal fieldwith center frequency vs and time-varying phase fs(t),and Er(t) is likewise defined as the reference field. In ex-periments that measure second-order correlation, n 5 2,the detected signal is proportional to the real part of thefunction
2002 Optical Society of America
Schmidt et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. B 1931
A2~tD! 5 E2`
`
@ uEs0~t !u4 1 uEr0~t 2 tD!u4 1 4uEs0~t !u2
3 uEr0~t 2 tD!u2#dt 1 2 exp~iv0tD!
3 E2`
`
@ uEs0~t !u2 1 uEr0~t 2 tD!u2#
3 Es0~t !Er0* ~t 2 tD! (2a)
3 exp$i@ fs~t ! 2 fr~t !#%dt 1 c.c. (2b)
1 exp~2iv0tD!E2`
`
Es02~t !~Er0* !2~t 2 tD!
3 exp$2i@ fs~t ! 2 fr~t !#%dt 1 c.c., (2c)
where the electric fields are assumed to be linearly polar-ized along the same direction in space and vs 5 vr5 v0 . Equation (2) indicates that A2(tD) comprisesthree frequency components, centered at zero, v0 , and2v0 , which can be extracted individually from the varia-tion of A2(tD) with tD by Fourier transformation.
The integral
Gn~tD! 5 E2`
`
@ f1~t !f2* ~t 2 tD!#ndt (3)
defines the nth-order correlation of the functions f1(t)and f2(t 2 t); line (c) of Eq. (2) shows that A2(tD) de-pends on the second-order intensity correlation G2(tD) ofthe signal Es
2(t) and time-delayed replica (Er* )2(t2 tD). Second-order autocorrelation measurementswith Es(t) 5 Er(t) remain a popular choice for ultrafastpulse characterization, because for n > 3 the contribu-tions of higher orders of Gn(tD) must be taken intoaccount.19–21 There is, however, an advantage to maxi-mizing the order n of an autocorrelation measurement.In the limit that n → `, Eq. (3) shows that, for Es(t)5 Er(t),
Gn~tD! ——→large n
Es2n~t ! ——→
n→`
ds~t !, (4)i.e., G(tD) starts to mimic the pulse intensity profile tothe nth power, which itself tends toward a d function intime as n → `. This behavior is displayed in Figs. 1(a)–1(c), which illustrate how the autocorrelation integralGn(tD) for a pair of identical Gaussian functions varieswith increasing n. For intermediate values of n, wherethe d-function limit is not achieved, a deconvolution pro-cedure is required for extraction of the pulse envelopeshape of the signal field. Inspection of Eq. (2) revealsthat An(tD) varies with n in a manner analogous to thatfor Gn(tD). Figures 1(d)–1(f ) indicate what happens foran interferometric record of An(tD). For n 5 2, the well-known second-order autocorrelation is obtained with the
Fig. 1. Autocorrelation behavior as a function of the nonlinear order coefficient n of the correlation integral. (a)–(c) Autocorrelationintegral Gn(tD) 5 *2`
` @Es(t)Er* (t 2 tD)#ndt as defined by Eq. (3) (solid curves), uEs(t)u2 (dashed curves), and uEs(t)u2n (dotted–dashedcurves) for the values of n indicated, where Eq(t) 5 Eq0(t)exp@ivqt 1 fq(t)# (q 5 s 5 r) are two identical E fields with Eq0(t)5 exp(2t2/4sq
2), vq 5 0.6 fs21, fq(t) 5 0 for all t, and sq 5 8.5 fs. The designation of time on the abscissa refers to tD for Gn(tD) andto t for uEs(t)u2 and uEs(t)u2n. (a) Shows that for n 5 2 the FWHM of G2(tD) is 21/2 times larger than the FWHM (tP 5 20 fs) ofuEs(t)u2 5 uEr(t)u2, as expected for Gaussian functions. (b), (c) Show that Gn(tD) → uEs(t)u2n as n attains large values. (d)–(f ) Illus-trate An(tD) defined by Eq. (1) (solid curves), uEs(t)u2 (dashed curves), and uEs(t)u2n (dotted–dashed curves) for values of n as indicated,where Es(t) and Er(t) have the forms, and the abscissa has the meaning, given above. (d) Familiar second-order autocorrelation in-terferogram for n 5 2. (e), (f ) Show how An(tD) tends in the limit n → ` to a single interferometric peak with a FWHM approachingthat of uEs(t)u2n. [For the arbitrary values of vq , fq(t), and sq chosen here, An(tD) is dominated by a single peak, with two sidemaxima at tD 5 61.7 fs reaching 6% of the amplitude of the center, when n 5 200.]
1932 J. Opt. Soc. Am. B/Vol. 19, No. 8 /August 2002 Schmidt et al.
familiar ratio of 8:1 between maximum signal at totalpulse overlap (tD 5 0)and minimum signal at large posi-tive or negative time delay (tD → 6`). As n is in-creased, the number of interferometric peaks decreasesuntil, for a sufficiently high value of n that depends oncarrier frequency v0 and the time-dependent phase of theidentical (Gaussian) functions in the integrand, only thecentermost peak, with a full width approaching that ofthe signal field to the power 2n, is contained within theinterferogram of An(tD). In the limit n → `, therefore,measurement of an intensity autocorrelation gives di-rectly the duration of the signal field to nth order that isresponsible for the phenomenon of interest.
The autocorrelation technique discussed here takes ad-vantage of the ac-Stark shift of a molecular absorptiontransition,26 off-resonantly induced by a pair of ultrafastlaser pulses, to make interferometric autocorrelationmeasurements of the incident light. The A 2S1 n8 5 2← X 2Pr n9 5 0 two-photon Bohr resonance of nitric ox-ide (NO) at lR 5 409.8 nm (\vR 5 3.026 eV) excited by400-nm ultrafast laser pulses serves as the vehicle for theapplication given here, with detection of A 2S1 n8 5 2→ X 2Pr n9 5 2 fluorescence emitted when the Bohr fre-quency of the absorption temporally matches the appliedtwo-photon frequency used to map out the autocorrelationfunction. Figure 2(a) presents the relevant energy levelsand photon transitions. The highly nonlinear depen-dence of the Stark-induced fluorescence on laser intensityof the form SF@I(t)# } I0
n (where n varies with I0 overdifferent intensity ranges)27 results in an autocorrelationtrace that begins to resemble the nth power of the laserpulse: this gives a direct measurement of the light inten-sity that brings about the observed Stark shift; its down-side, from a practical viewpoint, is that increasing n alsoamplifies pulse-to-pulse energy instabilities and the pulsewidth must be extracted through a theoretical analysis ofelectron dynamics in a pair of intense oscillating fields.The ac-Stark autocorrelator described here can measurethe duration of laser pulses at peak intensities I0< 50 TW cm22 (and could be extended to higher ranges,
depending on the target gas), and there are, in principle,no optical limits to the central wavelength of pulses thatcan be characterized. The procedure can equally well beapplied to atoms and other molecules in the gas phaseand configured with different detection techniques; it isenvisaged that the approach may serve as a pulse mea-surement technique in extreme situations of high field in-tensity or short wavelength for which methods based onsolid-state materials are not so readily available.
In Section 2 we describe the experimental approachadopted to record interferometric autocorrelation spectrabased on the ac-Stark effect. Experimental results arepresented in Section 3. A theoretical model based on asemiclassical calculation of the light–matter interactionthat enables the experimental results to be understood interms of Rabi population cycling and dynamic Stark shift-ing of a two-level system is outlined in Section 4: Theo-retical calculations presented in Section 5 illustrate howthe pulse duration of the incident field can be unfoldedfrom the highly nonlinear molecular response measuredin the laboratory. Section 6 concludes the paper and listsadvantages and limitations of an ac-Stark autocorrelator.
Our approach is similar to that of Kobayashi and co-workers, who measured the autocorrelation widths of theseventh and ninth harmonics of a Ti:sapphire laser beamgenerated by nonlinear ionization of He and Ar and ex-plained their results by a quantum-dynamical treatmentof the laser–atom interaction in space–time.13,14
2. EXPERIMENTFigure 2(b) shows a schematic of the optical arrangementthat was used to make autocorrelation measurements ofultrafast laser pulses at 400-nm wavelength and .10-TWcm22 intensities by the dynamic Stark effect. Femtosec-
Fig. 2. (a) Electronic and vibrational energy levels of NO thatpertain to this research as a function of internuclear coordinate.Off-resonant two-photon excitation of the A 2S1 n8 5 2← X 2Pr n9 5 0 Bohr transition at l 5 400 nm (\vR5 3.026 eV) leads to dynamic Stark shifting of the optically con-nected levels, which is monitored in real time by detection ofA 2S1 n8 5 2 → X 2Pr n9 5 2 g-band fluorescence at l5 222 nm (shown as a zigzag arrow). The differences betweentwice the energy supplied by the incoming photons, representedby the vertical arrow, and the n8 5 2 vibrational level of theA 2S1 state, and that between the n9 5 0 and the n9 5 2 vibra-tional levels of the X 2Pr state, are exaggerated for clarity. Thepotential curve labeled X 1S1 represents the ground state ofNO1. (b) Schematic diagram of the optical arrangement of anac-Stark autocorrelator. Ultrafast light pulses from an ampli-fied Ti:sapphire laser at l 5 800 nm are frequency doubled bytype I phase matching in a 0.5-mm BBO crystal. The second-harmonic beam is directed via a Michelson interferometer beforecollinear focusing at one side (to minimize fluorescence quench-ing) in a quartz cell containing a slow flow of NO gas. The l/2plate and the polarizer control the energy of input pulses. De-tector D represents the combination of a monochromator and aphotomultiplier tube for collection of fluorescence emitted per-pendicular to the direction of propagation of the laser beam.
Schmidt et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. B 1933
ond laser pulses at 800 nm were obtained by use of a re-generatively amplified Ti:sapphire laser operating at 10Hz with typical energies of 2.0 mJ per pulse. To generate400-nm pulses the laser beam was directed through a 0.5mm b-barium borate (BBO) crystal cut for type I SHG at800 nm. Care was taken to minimize pulse-to-pulse en-ergy fluctuations caused by the high-order nonlinearity ofthe Stark-effect measurements. A half-wave plate placedbefore the BBO crystal served to reduce the efficiency ofSHG for selection of frequency-doubled pulse energies.In this study, input energies of eP 5 40–140 mJ per pulsewere found to be convenient, delivered with a maximumuncertainty of 65% across the range. The second-harmonic beam was routed via a standard Michelson in-terferometer arrangement as shown in Fig. 2(b), and therecombined beams were collinearly focused by a thinfused-silica lens of focal length 200 mm into a slow flow ofNO gas in a quartz cell. Fluorescence due to the A 2S1
n8 5 2 → X 2Pr n9 5 2 vibronic transition at l5 222 nm was collimated and then imaged by a thick
fused-silica lens (focal length, 15 mm) placed exterior tothe cell onto 1.2-mm-wide slits of a monochromator. (Inthese experiments the transitions to the two spin–orbitcomponents of the X 2Pr state were not resolved.) Lightdetection was accomplished by a photomultiplier tube.For a given time delay between the two laser pulses, theresulting signal, representing the decay of the dynami-cally Stark shifted n8 5 2 A 2S1 NO level over a timescale of 180–195 ns28,29 (determined by spontaneousemission to all Condon-accessible lower levels) was aver-aged over 100 laser shots by a digital oscilloscope beforebeing downloaded and integrated on a PC. Autocorrela-tion traces were obtained by advancing the optical pathlength of one arm of the Michelson interferometer withrespect to the other by a piezo-controlled translationstage with a distance resolution of 650 nm. The piezovoltage was controlled by the output BNC of the oscillo-scope, which in turn was controlled by the PC. This pro-cedure was repeated for time delays spanning the baseduration (.300 fs) of the autocorrelation trace obtainedover the 50-mm maximum travel of the piezo controller.
Of critical importance to a quantitative analysis ofStark-shifted fluorescence is characterization of the pulseenergy and spatial properties of the incident light. En-ergy measurements of the frequency-doubled pulses weremade before the place where the recombined beam en-tered the slow flow of gas. The transverse intensity dis-tribution was determined by capturing beam profiles on aCCD camera located at the focus of the frequency-doubledbeam. A Gaussian fit to the measured profiles gave a fullwidth at half-maximum (FWHM) beam waist of w(z0)5 50 mm, commensurate with a confocal parameter of.39 mm for the focused second-harmonic beam. For thedetection geometry employed in this study [Fig. 2(b)], onlyfluorescence emanating from a region 175 mm on eitherside of waist position z0 was monitored, thereby permit-ting the assumption of a constant intensity distribution ofsecond-harmonic light along the direction of propagationz, as monitored in the perpendicular direction by themonochromator.
A slow flow of NO was passed through a quartz cell at apressure of approximately 0.5 Torr maintained by a ro-
tary pump. Pressure was measured downstream of thecell by a Pirani gauge and was regulated by a needle valveon the upstream side. The operating pressure of NO waschosen to give the best signal without inducing significantfluorescence-lifetime shortening through collisions.28,29
3. EXPERIMENTAL RESULTSThe observed nonlinear response of NO A 2S1 n8 5 2→ X 2Pr n9 5 2 fluorescence as a function of applied la-ser intensity is depicted in Fig. 3(a). It can be seen thatfor the present experimental arrangement the fluores-cence signal SF@I(t)# is small until a total incident energyof the combined pulses of eP . 40 mJ per pulse isachieved, after which the fluorescence increases sharply.This behavior may be interpreted as a dynamic Starkshift of the A 2S1 n8 5 2 ← X 2Pr n9 5 0 resonance towithin the bandwidth of the two-photon excitation fre-quency of the incident-light field.30 Taking logarithms ofboth variables reveals that the slope of Fig. 3(a) changesfrom 1.93 6 0.14 below eP 5 50 mJ per pulse to 4.086 0.20 at higher pulse energies. The squared depen-dence at low eP values is consistent with off-resonant two-
Fig. 3. (a) Experimental NO A 2S1 n8 5 2 → X 2Pr n9 5 2fluorescence signal excited by ultrafast laser pulses at 400 nm asa function of pulse energy eP . The observed fluorescence exhib-its a threshold as the A 2S1 n8 5 2 ← X 2Pr n9 5 0 absorptionis shifted to within the two-photon bandwidth of the applied la-ser field, and it then increases nonlinearly, with a slight shoulderat eP . 170–180 mJ per pulse. (b) ua2(t0)u2, the calculated fluo-rescent population in the first excited state of an electron con-fined to a one-dimensional box, versus peak intensity of a Gauss-ian pulse with temporal duration (FWHM) tP 5 97 fs.
1934 J. Opt. Soc. Am. B/Vol. 19, No. 8 /August 2002 Schmidt et al.
photon excitation of the A 2S1 n8 5 2 ← X 2Pr n9 5 0transition. The result displayed in Fig. 3(a), however,shows evidence of an extra feature not observed in an ear-lier presentation of these curves30: the increase inSF@I(t)# with pulse energy exhibits a kink between eP. 170 and eP . 180 mJ incident pulse energies, which inSection 5 we attribute to Rabi cycling of population be-tween the A 2S1 n8 5 2 and X 2Pr n9 5 0 vibrationallevels of the excited and ground electronic states.
Interferometric autocorrelation traces are shown inFig. 4 for two pulse energies. The trace obtained at eP5 60 mJ per pulse, displayed in Fig. 4(a), reveals thebackground-free nature of the interferometric ac Starkautocorrelator at low incident energies. At higher pulseenergies, illustrated for eP 5 140 mJ per pulse in Fig.4(b), the trace broadens as a result of saturation of thetwo-photon absorption probability at peak intensitiesnear tD 5 0 while it retains an essentially background-free autocorrelation. If the intensity envelope of thetrace shown in Fig. 4(a) is fitted to a Gaussian curve, theFWHM is 76 6 6 fs. At eP 5 140 mJ the autocorrelationenvelope cannot readily be curve fitted to an obvious func-tional form, though it may be analyzed in terms of the
Fig. 4. Interferometric autocorrelation traces obtained from dy-namic Stark shifting of the NO A 2S1 n8 5 2 ← X 2Pr n9 5 0two-photon Bohr resonance by laser pulses at l 5 400 nm andtotal incident energies of (a) eP 5 60 and (b) eP 5 140 mJ. Theautocorrelation envelope in (a) is fitted to a Gaussian functioncharacterized by a FWHM of 76 6 6 fs; that in (b) is character-ized by an average time duration of 138 6 11 fs derived from thesecond moment of the fluorescence energy density*2`
` SF@I(t)#dt.
second moment of the temporal dependence of the fluores-cent energy density (see Section 5 below).
As the intensity of the recombined pulses is increased,the shapes of individual interferometric maxima change,as shown in Fig. 5. At eP 5 40 mJ, displayed in Fig. 5(a),each peak has a shark-tooth-shaped appearance becauseof the extreme nonlinearity of the dependence of A 2S1
n8 5 2 → X 2Pr n9 5 2 fluorescence on applied intensity.When the pulse energy is increased to 92 and 140 mJ,Figs. 5(b) and 5(c) show that the peak shape becomesmore sinusoidal as off-resonant excitation of the A 2S1
n8 5 2 ← X 2Pr n9 5 0 transition exhibits saturationand population is coherently cycled between the two lev-els. An apparent phase shift in fluorescence fringes thatis observed on increasing the applied intensity is a reflec-tion of the precision error associated with establishing theabsolute time delay in different experiments rather thana manifestation of the physics of the level shifts. At eP. 150 mJ, Fig. 3(a) indicates that SF@I(t)# tends towardsaturation with further increases in I0 : Nonlinear levelshifting is optimally achieved, therefore, when the recom-bined pulse energy incident upon NO lies between the ap-pearance threshold and the first inflection point of Fig.3(a) at eP . 175 mJ per pulse. The appearance of har-monics in the Fourier transforms of the traces given in
Fig. 5. Close-up of interferometric autocorrelation traces ob-tained with laser pulses at l 5 400 nm and total incident ener-gies of (a) eP 5 40, (b) eP 5 92, and (c) eP 5 140 mJ. At lowerenergies the trace exhibits peaks with a shark-tooth shape as thelaser pulses constructively interfere. At higher energies the in-dividual peaks become sinusoidal as a result of saturation due toRabi cycling of population between the A 2S1 n8 5 2 and theX 2Pr n9 5 0 levels at various instantaneous Stark shifts of thetwo-photon transition that optically connects them.
Schmidt et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. B 1935
Fig. 5 confirms that the most nonlinear autocorrelation isobtained at the lowest laser intensity, Fig. 5(a), as mightbe suspected from the shape of the peaks as a function oftD .
In principle, the empirical response of SF@I(t)# on peakintensity may be used to calculate an intensiometric or in-terferometric autocorrelation trace for an assumed pulseprofile. This is generally true for any autocorrelationmeasurement in which the signal is generated essentiallyinstantaneously. In this scheme, the form of an inter-ferometric autocorrelation trace may be investigated byexpressing SF@I(t)# in terms of an instantaneous inten-sity I(t) according to
dSF@I~t !# 5 f@I~t !#dt. (5)
The function f @I(t)# describes the empirical variation ofthe time-integrated fluorescence signal as a function of la-ser intensity. Based on the result of Fig. 3(a), the depen-dence of SF@I(t)# on I0 was fitted to three functionalforms of f @I(t)#: one for the energy range eP5 0 –48 mJ; another for the range eP 5 49–82 mJ; and afurther form for eP > 83 mJ. The two lower-energy func-tions were derived from linear fits to a log–log plot ofSF@I(t)# versus I0 , whereas the function for the highestenergy range was fitted to a cubic intensity dependenceacross the appropriate range. The lower two ranges re-flect the regimes where the light–molecule interaction ex-hibits a squared dependence on incident intensity andthen changes as the A 2S1 n8 5 2 ← X 2Pr n9 5 0 reso-nance is shifted to within the applied two-photon band-width.
Functional fits to the variation of SF@I(t)# with I0 wereattempted to deduce the envelope shape of I(t) with littlesuccess: A distinct background found in the modeled au-tocorrelation trace at eP 5 60 mJ was further amplified ateP 5 140 mJ, contradicting the behavior anticipated inFigs. 1(d)–1(f ) and observed by experiment as demon-strated in Fig. 4. Owing to this discrepancy, it was notpossible to estimate the pulse width with any useful ac-curacy by this approach. In addition, from an experi-mental perspective, it is difficult to measure a responsecurve such as Fig. 3(a) accurately at low energies, wherethe signal is small and susceptible to pulse-to-pulse en-ergy fluctuations and scattered light augments the ob-served signal. These reasons do not fully explain the in-ability of Eq. (5) to yield pulse widths based on the ac-Stark shift of an absorption resonance, however: thefundamental cause of the erroneous predictions of Eq. (5)is that the model itself is flawed.
It is not generally true that, for an ac-Stark-shiftedresonance between atomic or molecular levels, the effectof two time-separated pulses will be twice the effect of anyindividual pulse. For detection of A 2S1 → X 2Pr fluo-rescence, the signal evolves over a lifetime of 180–195 nsfor spontaneous emission from the A 2S1 n8 5 2level,28,29 and, because in a femtosecond autocorrelationmeasurement the pulses are separated by a time muchless than this, the second pulse may reduce the fluores-cence excited by the first while making its own contribu-tion to the measured signal. This effect may be inter-preted in terms of a narrowing of the spectrumexperienced by an atom or a molecule when it is irradi-
ated by the second pulse, and it manifests itself as a re-duced background level, which is difficult to model interms of a simple time integration that neglects coherentinteractions. These phenomena are best addressed by di-rect solution of the time-dependent Schrodinger equation,as discussed in the following sections.
4. THEORYThe optical response of an atom or a molecule in a strong,oscillating electric field may be calculated by integrationof the time-dependent Schrodinger equation
id
dtuC~t !& 5 H~t !uC~t !&. (6)
In Eq. (6) and herein we take \ 5 1. Because the Hamil-tonian H(t) is a periodic function of time, a convenientapproach to solving Eq. (6) is to invoke Floquet’s theoremto express the state vector uC(t)& of the light–molecule in-teraction as a series of time-dependent probabilities thatrepresent the transient amplitude in each of the field-dressed eigenstates of H(t).31 In this paper we adopt atime-dependent Floquet approach to demonstrate howpulse widths can be extracted from data of the type shownin Fig. 4. In these calculations Eq. (6) is solved for anelectron in a one-dimensional box subject to an intense os-cillating field: the principal aim is not so much to make ahighly accurate estimate of the incident-pulse durationsbut rather to illustrate the procedure for determiningpulse widths from measurements of the ac-Stark effect.A detailed theoretical description of the interaction of in-tense laser light with NO has been given elsewhere.32
For any instant of time t, the state uC(t)& is projectedonto a set of orthonormal quasi-eigenstates $uf i(t)&% ofthe Floquet Hamiltonian that are themselves necessarilyfunctions of time. Equation (6) may then be written as
id
dtuC~t !& 5 H~t !(
ici~t !uf i~t !& 5 (
ie i~t !ci~t !uf i~t !&.
(7)
For the purposes of a numerical computation it is moreconvenient to express the dressed states in terms of a ba-sis set of unperturbed levels $u c j&% and to calculate thetime evolution of uC(t)& by means of the coefficients$aj(t)% of that basis. The time dependence of the $aj(t)%may then be obtained from a projection onto the quasi-eigenstates, giving
id
dtuC~t !& 5 H~t !(
jaj~t !u c j&
5 (i
(j
e i~t !aj~t !^f i~t !u c j&uf i~t !& (8)
on substitution into Eq. (6). The temporal propagation ofcoefficients $aj(t)% from t 5 2` to t 5 1t0 becomes
ak~t0! 5 ^ ckuC~t0!&
5 2i^ cku E2`
t0
dt (i, j
e i~t !aj~t !^f i~t !u c j&uf i~t !&.
(9)
1936 J. Opt. Soc. Am. B/Vol. 19, No. 8 /August 2002 Schmidt et al.
The information required for evaluation of Eq. (9) lies inknowledge of the quasi-eigenstates $uf i(t)&%, their eigen-values $e i(t)%, and how they vary with time. One canreadily calculate this information by diagonalizing H(t)in the time-independent basis set $u c j&%.
To exemplify the process of pulse width determinationmodel calculations were performed to mimic the responseof a quantum system to a nonperturbative oscillatingelectric field. The $aj(t)% were propagated in time ac-cording to Eq. (9) under the action of a time-dependentHamiltonian of the form
H~t ! 5 T 1 V 1 E~t !~x 2 L/2!, (10)
where V(x) 5 0 across the range 0 , x , L and V(x)5 ` at x < 0 and x > L outside a one-dimensional boxin which a single active electron was constrained to move.The time-dependent field of two identical interferingpulses of FWHM tP , one delayed with respect to the otherby tD , is represented classically by
E~t ! 5 E0$exp@24 ln~2 !t2/tP2 #cos~v0t !
1 exp@24 ln~2 !~t 2 tD!2/tP2 #cos@v0~t 2 tD!#%.
(11)
To mimic the experiments described in Section 3, the sym-metry of the box was broken by imposing a constant fieldD/L across its length to permit a two-photon transitionbetween the ground and the first excited states. In NO,the two-photon transition between the X 2Pr groundstate and the A 2S1 excited state is allowed because ofconsiderations of angular momentum. Box parameterswere chosen to imitate the molecular absorption: a valueof L 5 8.16 a0 reflects the A 2S1 n8 5 2 ← X 2Pr n95 0 two-photon transition energy, giving \vR . 48 800
cm21, compared with the spectroscopic value of 48 852cm21.33 An empirical value of D 5 0.54 eV was selectedsuch that the spectral line that arose from the two-photontransition between the ground and the first excited stateswas of comparable intensity to the one-photon transitionat intensities that correspond to the experimental condi-tions. The unperturbed basis set levels $ c j% are given by
u c j& 5 HA2/L sin~ jpx/L ! 0 , x , L
0 elsewhere, (12)
with boundary conditions a0(2t0) 5 1 and ajÞ0(2t0)5 0.
The Hamiltonian was diagonalized at discretized fieldstrengths across the range from 0 to 6E0 , and the timedependence of the quasi-eigenstates $uf i(t)&% and thequasi-eigenvalues $e i(t)% were calculated by interpolationsuch that convergence of the calculated data was obtainedwith respect to the spacing in E(t) between adjacent di-agonalizations. Convergence was achieved by use of atime step of 0.2 a.u. (5 as) and a basis set that comprisedthe five lowest-energy states $u c j&% of the unperturbedbox. Time integration was carried out for times fromt 5 250,000 a.u. (2`) to t 5 150,000 a.u. (t0), withpulse delay times tD fitting easily within this range. Theduration tP of the two pulses was chosen to be 4000 a.u.,equivalent to 97 fs. The fluorescence signal was taken as
the square modulus of coefficient ua2(t0)u2 of u c2& at timet 5 t0 after the end of the second pulse.
5. THEORETICAL RESULTS ANDDISCUSSIONIn this section the dependence of the fluorescence inten-sity is presented on different field parameters to examinethe prospects for autocorrelation measurements based ondynamic light shifts of molecular (or atomic) levels in anintense laser field. Figure 3(b) shows the variation ofua2(t0)u2, which represents the time-integrated fluores-cence signal SF@I(t)#, as a function of the peak intensityI0 of a pulsed field for two-photon excitation from theground to the first excited state of a one-dimensional box.Noticeable in comparison to the experimental resultshown in Fig. 3(a) is the presence of a shoulder beginningat I0 . 30 TW cm22, with further plateaus centered atI0 . 42, 50, 58 TW cm22. It is postulated that the step-wise increase in ua2(t0)u2 as a function of I0 arises fromRabi cycling of population between the optically con-nected levels of the box at different instantaneous Starkshifts of the same levels with respect to the photon en-ergy. As the field intensity develops during the pulseevolution, peaks of maximum population in the upperlevel pass consecutively through the dynamic two-photonresonance with the result that, in the time-integratedframe, the fluorescence intensity increases as the upperlevel is brought into range of potential Bohr resonancescontained within the spectral bandwidth of the pulse;ua2(t0)u2 then decreases (nominally to zero) as the popu-lation is coherently returned to the lower level by Rabi cy-cling. Experimentally [Fig. 3(a)], the effect is tentativelyobservable at eP 5 170–180 mJ per pulse, before otherphenomena such as ionization dominate the molecular re-sponse at field strengths near the pulse maximum. Thatthe observed fluorescence SF@I(t)# never returns to zeroat suprathreshold intensities is a result of the transversespatial intensity variation of the incoming laser beam:as the light field shifts the A 2S1 n8 5 2 ← X 2Pr n95 0 resonance beyond reach of the two-photon energy ata particular coordinate in space–time, say, at the pulsecenter, the contribution to fluorescence from that locationis necessarily diminished; this decrease, however, is coun-teracted by contributions from regions of space mappedout by lower-intensity portions of the laser beam on eitherside of the central propagation axis. The fluorescencesignal SF@I(t)# therefore tends to a series of limiting val-ues before it reaches a final plateau. Support for this in-terpretation is provided by calculations of the spectralvariation of fluorescence at different field strengths, asdiscussed in more detail elsewhere.32
Owing to the similarity between the experimental andthe theoretical response curves of Fig. 3, a theoreticalpeak intensity may be chosen at which to simulate an au-tocorrelation trace. In Fig. 3(a) a pulse energy of eP5 100 mJ per pulse is equivalent to a peak intensity ofI0 5 25 TW cm22 on the theoretical plot of Fig. 3(b). Us-ing this ratio, the observed autocorrelation trace shown inFig. 4(a) can be calculated for a peak intensity, at maxi-mum overlap in space–time of the two pulses, of I05 15 TW cm22 (corresponding to two time-separated
Schmidt et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. B 1937
pulses, each with a quarter of that intensity). Likewise,the autocorrelation trace obtained at eP 5 140 mJ perpulse shown in Fig. 4(b) may be simulated by use of acombined pulse peak intensity of I0 5 35 TW cm22.
Calculated autocorrelation traces are presented in Fig.6: in both diagrams (a) and (b) it can be seen that thebackground-free nature of the experimental measure-ments is reproduced. The Gaussian envelope of Fig. 6(a)is found to have a FWHM of 50 fs, much lower than theduration of an individual pulse, set to 4000 a.u. (97 fs) inthe calculations. From the ratio of full widths at half-maxima of the lower-intensity experimental and calcu-lated autocorrelation traces [Figs. 4(a) and 6(a)], a pulsewidth of tP 5 147 6 12 fs may be extracted for the laserpulses at 400 nm used in the experiment. This resultwas validated by comparison with values of tP derivedfrom cross-correlation measurements of the second-harmonic and fundamental Ti:sapphire beams. Thesevalidations were carried out by type I third-harmonic gen-eration in a 100-mm-thick BBO crystal and gave the re-sult tP 5 140 6 10 fs for second-harmonic pulses witheP 5 60 mJ, assuming a Gaussian envelope for the E-field. The shape of the computed autocorrelation trace
Fig. 6. Calculated interferometric autocorrelation traces de-rived from dynamic Stark shifting of NO A 2S1 n8 5 2← X 2Pr n9 5 0 for identical Gaussian pulses with tP 5 97 fsand peak intensities at a maximum space–time overlap of (a)I0 5 15 and (b) I0 5 35 TW cm22. The autocorrelation envelopein (a) is fitted to a Gaussian function with a FWHM of 50 fs; thatin (b) is characterized by an average time duration of 77 fs de-rived from the second moment of the fluorescence energy density*2`
` SF@I(t)#dt.
shown in Fig. 6(b) for the higher intensity of I05 35 TW cm22 is, like its experimental counterpartgiven in Fig. 4(b), not amenable to curve fitting to a stan-dard functional form. Instead, an analysis based on theaverage of the second moment of the temporal depen-dence of the fluorescent energy density34 gives values of138 6 11 and 77 fs, respectively, for the widths of auto-correlation envelopes of Figs. 4(b) and 6(b). From the ra-tio of these values, a duration of tP 5 174 6 14 fs may bedetermined for the laser pulses that we used to obtain theexperimental measurement, 18% larger than the resultobtained from the lower-intensity autocorrelation of Fig.4(a). Figure 7 shows high-resolution autocorrelationpeaks calculated at values of I0 chosen to match thosethat were used to obtain the corresponding experimentaldata shown in Fig. 5. For the reasons discussed in Sec-tion 3, the individual peaks of the interferometric autocor-relation trace change appearance from shark-toothed tosinusoidal shapes as the applied field strength is in-creased.
6. CONCLUSIONS AND FUTURE WORKIt has been demonstrated that the ac-Stark effect associ-ated with a molecular absorption resonance affords ameans for making background-free, highly nonlinear au-tocorrelation measurements of ultrafast laser pulses.The viability of an ac Stark autocorrelator has been illus-trated by taking advantage of the light shift of the A 2S1
Fig. 7. Calculated interferometric autocorrelation traces foridentical Gaussian pulses with tP 5 97 fs and peak intensities ata maximum space–time overlap of (a) I0 5 1.0, (b) I0 5 2.8, and(c) 4.3 TW cm22.
1938 J. Opt. Soc. Am. B/Vol. 19, No. 8 /August 2002 Schmidt et al.
n8 5 2 ← X 2Pr n9 5 0 two-photon resonance of NO tocharacterize the widths of intense laser pulses at 400 nm.An analysis based on integration of the time-dependentSchrodinger equation has been adopted to calculate theinterference fringes that result from the interaction oftwo classical light fields with an electron confined to os-cillate in a model one-dimensional box. From simula-tions of this kind, a determination of the pulse durationcan be extracted from experimental data if a specific formfor the pulse envelope shape is assumed. The approachcan be refined to make more-accurate measurements ofpulse widths by use of a more realistic set of atomic or mo-lecular parameters to mimic the level shifts generated bythe applied light fields. The calculations further permitthe variation of upper-state population with incident la-ser intensity to be assessed, generating in this case aquantitative estimate of the magnitude of the ac Starkshift that accompanies the NO A 2S1 n8 5 2 ← X 2Prn9 5 0 two-photon resonance30 and highlighting the ef-fect of coherent Rabi cycling between the two opticallyconnected levels.
An autocorrelation measurement based on the ac-Starkeffect suffers from the same disadvantages as any otherautocorrelation technique, namely, that it necessitates re-course to a decorrelation procedure to extract informationon the pulse envelope and the optical phase and how itvaries with time.24 In the particular application de-scribed here, the approach would be susceptible to exces-sive pulse-to-pulse energy fluctuations due to the nonlin-ear dependence of the dynamic Stark effect on incidentintensity; it also requires a solution to the time-dependent Schrodinger equation to obtain a measure ofthe pulse width. Nevertheless, the highly nonlinear de-pendence of the fluorescence signal on applied intensitydoes offer some distinct advantages: In the limit thatn → `, the autocorrelation width becomes the same asthe effective width of the pulse intensity envelope to thepower n that gives rise to the induced light shift [see re-lation (4)], thereby providing an immediate measure ofthe available time resolution associated with observationsof the dynamic Stark effect. Increasing the order of anautocorrelation measurement introduces the further,practical advantage that a shorter range of time delaysneed be scanned to map out An(tD).
The ac-Stark autocorrelator is easy to implement andmay be combined with highly sensitive detection schemes.There would appear, at least in principle, to be no mini-mum or maximum limit to the photon frequency to whichit could be applied, provided that the degree of detuningfrom Bohr resonance v0 2 vR and applied intensity canoff-resonantly populate an atomic or molecular level bydynamic light shifting. These factors suggest that an au-tocorrelator based on the ac-Stark effect could be usefullyemployed to measure the duration of pulses constructedfrom submillimeter or infrared waves35 on the one handand of vacuum-ultraviolet pulses on the other.22,23,36
With available femtosecond laser technology, we suspectthat a fairly immediate extension of this approach couldbe made to characterize ultraviolet and shorter-wavelength pulses by use of transient resonances be-tween high-lying atomic levels shifted by amounts close tothe ponderomotive energy.37 The approach could
therefore be adopted to measure pulses produced by high-harmonic generation (HHG) in atomic gases,38 althoughunlike other proposals,39,40 it does not offer an immediatedetermination of the carrier phase.
The dynamic range of an ac-Stark autocorrelator is nec-essarily set by the carrier frequency, pulse duration, andpeak intensity of the incident field and the level structureof the target atom or molecule, in addition to consider-ations of signal detectability and pulse stability. In con-trast to other methods,41,42 a massive dynamic range isnot, by the very nature of the light–matter interaction, anintrinsic feature of measurements restricted to one atomor molecule, though a judicious choice of field–atom (mol-ecule) parameters can allow for a larger dynamic rangethan that utilized here.
In the experiments reported in this paper the range ofinput intensities was ultimately limited by the BBO crys-tal that was used to generate frequency-doubled Ti:sap-phire light and the collection of sufficient fluorescence in-tensity. There is, however, a restriction on the maximumintensity and minimum pulse width that can be measuredby ac-Stark shifting of levels that is inherent in the phe-nomenon. As the bandwidth of the incoming pulse is in-creased, more than one upper level may be populated, andan analysis of the time dependence then has to incorpo-rate wave-packet motion37 including, for short pulses orclosely spaced (Rydberg) levels, the coherent dynamics ofdifferent electronic states of an atom or molecule.43,44
When the pulse width ultimately becomes so short or thepeak intensity so high that ac-Stark broadening of a lineexceeds the unperturbed eigenvalue, the concept of aStark shift itself breaks down26 and the analysis of Sec-tion 4 necessarily fails. For extremely short pulses itwould also be necessary to take into account the disper-sion of the measuring device,45 in this case through thewavelength dependence of the refractive index of the gas-eous target;46 dispersive effects could, however, be mini-mized by performing spatiotemporal overlap of laserpulses in a jet expansion rather than in a slowly flowing(or static) gaseous target. In applications to HHG pulsemeasurement, theoretical research indicates that, at opti-cal frequencies, the intensity of the generating fieldshould be restricted to values below 1016 W cm22,47 be-cause at higher field strengths the magnetic componentcauses the ponderomotively oscillating electron to driftaway from the nuclear Coulomb potential, thereby limit-ing the intensity of HHG light. Recent calculationsbased on a hydrodynamic formulation of the electric di-pole of the entire electron cloud surrounding an atomicnucleus suggest, however, that this limit may be circum-vented through the application of generating fields atphoton energies in the vacuum-ultraviolet range.48 Inthemselves, though, these considerations do not affect theapplicability of the method to measure HHG pulses of at-tosecond duration, provided that they can be detected anddo not bring about overwhelming Stark broadening forthe reasons outlined above. The method demonstratedhere could serve as an alternative to two-color atomicphotoionization, invoked recently as a means with whichto characterize coherent high-harmonic light with 220-aspulse widths.49 In conclusion, we should like to suggestthe present approach as a possible addition to the tech-
Schmidt et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. B 1939
niques for measuring attosecond pulses suggested byConstant et al.50
ACKNOWLEDGMENTST. W. Schmidt thanks Churchill College Cambridge for aresearch studentship and the University of Sydney for anEleanora Sophia Wood Travelling Scholarship. R. B.Lopez-Martens acknowledges the award of a research stu-dentship by the Engineering and Physical Sciences Re-search Council (EPSRC; UK). This study was supportedby the EPSRC, the Isaac Newton Trust, and the Royal So-ciety of London through generous equipment grants.
*Present address; Department of Chemistry, Massa-chusetts Institute of Technology, Cambridge, Massachu-setts 02139.
†Present address, Department of Physics, University ofNewcastle, Newcastle upon Tyne NE1 7RU, England.
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27. We write the fluorescence signal as SF@I(t)#, as opposed toSF@I0#, to emphasize excitation of fluorescent populationthroughout the entire pulse duration and to indicate thepossibility of extracting I(t) from SF@I(t)#, as discussed inSection 3 below.
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