DOI: 10.1007/s00340-006-2549-7

Appl. Phys. B (2007)

Lasers and OpticsApplied Physics B

j.c. delagnesa. monmayrantp. zaharieva. arbouetb. chatelb. girardm.a. bouchene�

Compensation of resonant atomic dispersionusing a pulse shaperLaboratoire Collisions, Agrégats, Réactivité (UMR 5589, CNRS – Université Paul Sabatier Toulouse 3),IRSAMC, 31092 Toulouse Cedex 9, France

Received: 20 July 2006/Revised version: 21 November 2006© Springer-Verlag 2007

ABSTRACT An ultra-short pulse propagating in a resonantdense atomic medium experiences an important distortion dueto a strong modification of its spectral phase. This distortioncannot be corrected using the usual simple dispersive devices(a pair of prisms, gratings, etc.). We present here an experi-mental demonstration of the compensation of this effect usinga dual 640-pixel high-resolution pulse-shaper device. A cross-correlation intensity measurement combined with the XFROG(cross-correlated frequency-resolved optical gating) spectralphase measurement of the compensated pulse are performed;efficient correction of the resonant dispersive phase is shown.A spectacular temporal compression of the propagating pulse isthen obtained.

PACS 92.60.Ta; 42.50.Md; 42.65.Re; 42.79.-e

1 Introduction

Ultra-short optical pulses generally experiencestrong distortion when they propagate through dispersive me-dia. In the weak-field regime, these dispersion effects areentirely contained in the frequency dependence of the spectralphase ϕ(ω) characterizing the transmittance of the medium.For laser frequencies far from any resonance, the mediumis transparent and ϕ(ω) is a slowly varying function whichcan be approximated by its polynomial expansion. The first-and second-order derivatives are the group delay and group-velocity dispersion (GVD), respectively. The GVD inducesa linear variation of the instantaneous frequency (chirp) anda temporal broadening of the pulse [1]. Compensation of thiseffect can be achieved with a pair of diffraction gratings orprisms with opposite GVDs [2, 3]. On the other hand, thelaser pulse is severely distorted when propagating througha medium with absorption resonances lying within the spec-tral bandwidth [4]. For instance, this situation occurs whena femtosecond pulse propagates in the atmosphere. The dis-persion induced by the water vapour is then mainly respon-sible for the deformation of the pulse [5]. Since low order

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polynomial expansion of the spectral phase is no longer valid,the above-mentioned compressing devices are useless. Propa-gation of a chirped pulse through a resonant medium resultsin strong modulations due to interferences between the inci-dent and the radiated fields [6]. Distortion-free propagationis obtained for an incident field without any frequency com-ponents in the spectral region where dispersion occurs. Forinstance, this is the case of a train of ultra-short pulses wheretwo adjacent pulses are locked with a relative phase of π atthe resonance frequency. As a result, the pulse train propa-gates with very low distortion [7, 8]. For different complexsequences, or a single pulse, compensation of resonant dis-persion can obviously be achieved only with optical devicesthat can produce rapid variations of the spectral phase, suchas pulse shapers [9]. They were successfully implementedin various fields such as coherent control [10, 11], quantumstate and ultra-short electric field reconstruction [12–14] andoptical communications [15].

In this paper, we show how the dispersion introduced bya resonant atomic medium on an ultra-short pulse can be com-pensated using such pulse-shaper devices. The feasibility isexperimentally demonstrated for considerably high opticaldepths (α0� ∼ 21 000; α0 is the absorption coefficient at reson-ance and � the length of the medium). Finally, we discuss thelimitations of such a method.

2 Basics of ultra-short-pulse propagation

The temporal distortion of a Fourier-transformlimited ultra-short pulse is constrained in the weak-fieldregime by two propagation laws [16]. First, when the pulse isshort, its spectral width is larger than the absorption line andso most of the initial energy is transmitted. Dispersion effectsare prevailing and the quantity

W(z) =+∞∫

−∞ε2 (t, z) dt,

proportional to the pulse energy, is then constant (ε (t, z) be-ing the pulse envelope). The second propagation law is theMcCall–Hahn area theorem which states that the pulse area

Applied Physics B – Lasers and Optics

decreases exponentially with the propagation distance z as

+∞∫

−∞ε (t, z) dt = e−α0z

+∞∫

−∞ε (t, 0) dt . (1)

The pulse envelope thus develops an oscillatory temporalstructure to satisfy these two conditions. The transmitted fieldafter a length � can be expressed as

ε (t, �) = 1√2π

+∞∫

−∞ε̃ (ω, 0) T (ω, �) e−iωt dω , (2)

where ε̃ (ω, 0) is the Fourier transform of the incident pulse:

ε̃ (ω, 0) = 1√2π

+∞∫

−∞ε (t, 0) eiωt dt (3)

and T (ω, �) = eiϕ(ω,�) is the spectral field transmission of theatomic medium with

ϕ(ω, �) = −ω�c

(n(ω)−1) � − α0�∆dω−ω0 (4)

FIGURE 1 Spectral behaviour of the laser pulse: (a) sine of the phase andamplitude transmitted by the atomic medium alone in the condition of ourexperiment (theoretical simulations), (b) sine function of the phase shift in-troduced by the pulse shaper, (c) spectrum intensity measured at the entrance(dotted line) and exit (solid line) of the pulse shaper

(n is the refraction index and ∆d the Doppler width). In-deed, each frequency component ω experiences a phaseshift which is significant (ϕ ≥ 1) over a spectral range[ ω0 −α0�∆d, ω0 +α0�∆d ]. The real and imaginary parts ofthe spectrum exhibit strong oscillations for frequencies closeto the atomic resonance (Fig. 1a). In a simplified view of theprocess, the rapid oscillations of the phase give in the temporaldomain small contributions that spread over a large temporalinterval. The two side lobes separated by the spectral quan-tity 2α0�∆d lead to the main contributions, which result frombeatings between these two frequency bands. Direct com-pensation of these dispersion effects is achieved with a pulseshaper introducing a spectral phase ϕs(ω) opposite to the oneintroduced by the medium (see Fig. 1b).

3 Experimental method

The experimental demonstration of this effect isperformed in atomic rubidium, on the 5s2S1/2 → 5p2 P1/2transition at λ0 = 794.76 nm. A home-built Ti:sapphire fem-tosecond oscillator provides optical pulses with about 150-fsduration (full width at half maximum (FWHM) of the inten-sity) and a spectrum bandwidth (FWHM) of ∆L = 23 rad THz(7.7 nm). A small fraction of the beam is used as a refer-ence and the second part is successively sent into the pulseshaper along with a 12-cm-long heat pipe containing anatomic vapour of rubidium. The pulse shaper consists ofa zero-dispersion line (4- f set-up) composed of one paireach of reflective gratings and cylindrical mirrors [17, 18].Its active elements are constituted by a combination of two640-pixel liquid-crystal spatial light modulators (LC-SLM,Jenoptik) located in the common focal plane of both mir-rors of the 4- f line. Each pixel is 100-µm wide. The LC-SLM has a high spectral resolution in both phase and am-plitude. The spatial resolution of the pulse spectrum is δ =0.18 rad THz/pixel (0.06 nm/pixel) and corresponds to a suf-ficiently wide temporal range (35 ps at 795 nm). The tem-perature of the rubidium heat pipe is fixed at 160 ◦C forwhich the corresponding vapour pressure of rubidium is about8 ×10−3 mbar. Argon is also used as a buffer gas at a pressureof 20 mbar.

The transmitted pulse is measured via intensity cross cor-relation with the reference beam. Experimental results show-ing the temporal behaviour of the pulse are represented inFig. 2. In all cases, the dotted line represents the intensity pro-file of the laser pulse at the entrance of the medium and at theexit of the pulse shaper when the latter is not active. It repre-sents the intensity profile to recover after pre-compensationby the pulse shaper and propagation in the heat pipe. Theintensity cross correlation is 250 fs FWHM and does not co-incide with the Fourier-transform duration corresponding tothe spectral breadth. Results after propagation through theheat pipe are given in Fig. 2. In Fig. 2a, the pulse shaperis not active and thus no phase compensation is introduced.The transmitted pulse exhibits oscillations due to the atomicdispersion that distorts the pulse profile. The experimentalcurve is fitted by a theoretical curve with the adjusted param-eter α0�∆d|fit = 13.6 rad THz (4.55 nm) obtained by a leastsquares fit method, which is in agreement with the calculatedheat-pipe characteristics (giving α0�∆d|calc. = 13.4 rad THz).

DELAGNES et al. Compensation of resonant atomic dispersion using a pulse shaper

FIGURE 2 Temporal behaviour of the laser pulse: propagation through (a)atomic medium only, (b) pulse shaper only, (c) both of them. For each case,we represent the input pulse (dotted line, raw data), transmitted pulse (solidline, raw data) and theoretical fit (dashed line)

Figure 2b represents the temporal profile of the laser pulseafter propagation in the pulse-shaper device only. The phaseshift requested from the pulse shaper is ξ/(ω−ω′0) withξ = +12 rad THz and ω′0 � ω0 within the spectral accuracyof the spectrometer (∼ 0.3 rad THz or 0.1 nm). In compari-son with Fig. 2a, the phase inversion produces a time-reversedintensity profile. The experimental curve is fitted by a the-oretical curve taking into account the actual phase and am-plitude transmission of the pulse shaper and thus differingfrom the simple phase dependence requested. When the opti-cal pulse propagates through both the heat pipe and the pulseshaper, the dispersion effects cancel each other out and thedistortion of the transmitted pulse is significantly reduced(Fig. 2c, solid line). Note that the parameter ξ above was cho-sen experimentally to achieve the best compensation effect.It does not coincide rigorously with the value α0�∆d|fit =13.6 rad THz fitted from the best heat pipe data alone. Thedashed line in Fig. 2c represents the theoretical fit of the trans-mitted pulse taking into account both effects of the heat-pipeand the pulse shaper. This will be discussed below in moredetail.

4 XFROG measurement – phase reconstruction

The compensation considerably suppresses thelong ringing tail present after the pulses propagate throughthe atomic medium. Most of the dispersive spectral phase is

obviously compensated in this experiment since the retrievedpulse is close to the initial pulse. Nevertheless, we have per-formed XFROG (cross-correlated frequency-resolved opticalgating) [19] measurements in order to quantify this conclu-sion. The signal emitted from the BBO crystal in the phase-matching direction is then analysed with a fibre-coupled spec-trometer. The fibre is injected with a f = 5 cm focal lengthlens.

Figure 3 depicts the raw XFROG data traces. We repre-sent the XFROG signals of the output of (a) the pulse shaperprogrammed with the appropriate spectral phase for compen-sation, (b) the vapour cell alone and (c) the combination ofthe two, leading to compensation. Note the time inversion ofthe scale between Fig. 3a, b and c. In all cases (Fig. 3a to c),we have used a linear grey scale from 0 to 4500 with 512 lev-els. The three contour plots correspond respectively to 80, 260and 1350 of the signal intensity. In this representation, we cansee the correspondence between the lobes of each XFROGtrace associated with the pulse shaper (Fig. 3a) and the atomicvapour (Fig. 3b). Although the centre positions of each lobeshifts progressively in time, there is a clear suppression ofthese lobes on a long time scale in Fig. 3c. However a pedestalremains as observed in Figs. 2c, 3c and 5c.

This pedestal arises from the finite resolution of thepulse shaper, which cannot compensate the divergence ofthe dispersion at resonance. Moreover, the limited band passmostly affects the amplitude of transmitted spectral com-ponents even when a ‘phase-only’ filter is applied. Indeed,large phase steps (e.g. ‘π-jump’) between consecutive pix-els lead to spectral holes in the transmitted amplitude [20](see below). Before examining the spectral intensity of thetransmitted pulse, let us consider the spectral phase extractedfrom the XFROG signal. The PCGPA (principal compon-ent generalized projections algorithm) [21] method is usedto retrieve the complete set of amplitude and phase (As(t),ϕs(t)) and (Ag(t), ϕg(t)), where subscripts s and g denote the‘signal’ and ‘gate’, respectively. We resample the data intoa square 2N ×2N array (here N = 128). The correspondingtemporal and spectral resolutions are then 7.8 fs and 1.1 nm,respectively.

The spectral phases ϕs(ω) and ϕg(ω) are shown in Fig. 4.They are obtained by applying a fast Fourier transform (FFT)on the retrieved signal and gate. The two phases are repre-sented over a pulsation interval of ±21 rad THz correspond-ing to twice the FWHM of the spectral amplitude. Becausethe linear part does not affect the pulse shape, it has been sub-tracted from the spectral phases ϕs(ω) and ϕg(ω).

On the interval presented here, ϕg(ω) carries a phase fluc-tuation of 600 mrad (peak to peak). This important amount offluctuation on the gate spectral phase is due to the presenceof a noise on the associated temporal amplitude (not shownhere) retrieved by the PCGPA. Indeed, the iterative methodand convergence criteria of the reconstruction algorithm candistribute arbitrarily the noise between signal and gate [22]. Inour reconstruction, the noise has a bigger contribution on thegate rather than on the signal. As for ϕs(ω), the phase varia-tion is 60 mrad, which is over the range where the spectrumamplitude is significant. This value indicates that the phaseis sufficiently flat and compensated – within the resolution of1.1 nm – and it explains the excellent results of the compensa-

Applied Physics B – Lasers and Optics

FIGURE 3 Raw XFROG data: (a) XFROGtrace produced by applying a spectral phasewith the pulse shaper alone (ξ = 9.3 rad THz),(b) XFROG trace of the pulse transmitted bythe atomic medium alone in the condition ofour experiment ( α0�∆d|calc. = 10.5 rad THz),(c) XFROG trace of the compensated pulse(combining pulse shaper and atomic vapour withthe parameters of (a) and (b)). The differencebetween these values is discussed in the text

tion in the temporal domain, especially the width of the mainpeak.

5 Limitation and alternative

The discrepancy existing between actual and idealcompensations finds its origin in both the undersampling ofthe rapidly varying phase near resonance and the resultingmodification of the amplitude of the transmitted spectral com-ponents. The former is due to the finite resolution of thepulse shaper (0.06 nm per pixel), which is unable to repro-duce mask functions with characteristic scales of variationcomparable to or smaller than the pixel size. The latter isdue to the finite spot size of each monochromatic compon-ent as a result of diffraction [23]. In general, pulse shapers are

designed so that the spot size of each monochromatic com-ponent in the SLM plane is of the same order as the pixelsize1. In the case of a slowly varying function, these two ef-fects combine and achieve a smoothly varying function. Inthe opposite case, too rapid variation results in holes in thetransmitted amplitude1 [20]. Because of the finite resolution

1 The spot size is chosen to achieve a compromise between large spotswhich reduce the resolution and a smaller spot size that causes thepixellated structure of the mask to appear as defects such as temporalrebounds or pedestals. The smoothing due to diffraction regards the com-plex number associated with the electric field of each monochromaticcomponent, while the undersampling concerns the value of the phase.These two effects do not compensate for large phase steps. As an ex-ample, a phase step of π results in attenuated transmitted amplitude forthe components in the vicinity of the gap between the two adjacent pixels

DELAGNES et al. Compensation of resonant atomic dispersion using a pulse shaper

FIGURE 4 Spectral phases ϕs(ω) and ϕg(ω) extracted from FFT performedon the gate and signal retrieved using PCGPA on XFROG traces in the case ofcompensation (Fig. 3c): circles: gate spectral phase ϕg(ω); triangles: signalspectral phase ϕs(ω)

(including diffraction limits in the Fourier plane), whateverthe resolution of the device may be, there will always bea small spectral region where the phase dispersion cannot bereproduced correctly. Indeed, the oscillations are narrowerand narrower when approaching the resonance frequency (4),and the pulse shaper also significantly modifies the intensitynear phase jumps.

FIGURE 5 Compensation with a flat phase ϕs(ω) around the atomic res-onance: (a) sin ϕs(ω) introduced by the pulse shaper, (b) spectrum intensitymeasured at the exit of the pulse shaper, (c) initial (dotted line) and transmit-ted (solid line) pulses with theoretical fit (dashed line)

The joint effect of diffraction and pixellation is signifi-cant if the phase variation between adjacent pixels is ∆ϕn =ϕn+1 −ϕn ≥ 1. From (4), ∆ϕn � δα0�∆d (ω−ω0)−2 and istherefore significant in a spectral domain ∆pix = 2√(α0�∆d)δcentred around the atomic resonance ω0, within which thetransmitted electric field is strongly distorted. Introducing thequantity ∆comp = min (2α0�∆d,∆L) over which the compen-sation has to be done, one needs ∆pix ∆comp to reducethe influence of these limitations and to ensure that distor-tions remain small. In our case ∆pix = 3.1 rad THz (1 nm),∆comp = 23 rad THz, and then ∆pix/∆comp = 0.13 indicatingthat 13% of the laser bandwidth is affected.

These effects contribute to the discrepancy observed nearthe atomic resonance between the theoretical phase requestedand that restored by the pulse shaper (Fig. 1b), and the at-tenuation of amplitude as shown in the transmitted spectrum(Fig. 1c). Figure 1c displays the transmitted intensity spec-trum with the pulse shaper off (dotted line) or activated (solidline). The pulse shaper significantly modifies the intensitynear the atomic resonance2. Figure 1b and c also confirm thatthe spectral domain over which this effect occurs is of thesame order as ∆pix over which the phase variation is import-ant. These discrepancies explain why the best compensationis obtained for a value of ξ that is slightly different from thevalue of the optical thickness calculated from the experimen-tal parameters or deduced from the cross correlation in thenon-compensated case (Fig. 2a).

The strong attenuation resulting from this effect can beavoided by replacing the rapid phase variation with a con-stant value on an interval of the order of ∆pix (Fig. 5a). Inthis case, the amplitude transmission is greatly improved nearthe atomic resonance (Fig. 5b). The temporal behaviour ofthe transmitted pulse is shown in Fig. 5c. The initial pulse iswell restored and a similar quality of compression is obtainedwhen comparing to Fig. 2c. However, the two methods are notequivalent. First, at the exit of the atomic medium, the phase ofthe spectral components near the central wavelength is almostcompensated in the first case but not at all in the second. Sec-ondly, more energy from the concerned region is transmittedin the second case, since the transmission is not altered. Thus,in the time domain, more energy is expected to spread out inthe wings of the transmitted pulse. This effect is observed inFig. 5c, where a small peak appears near 600 fs.

6 Conclusion

We have shown that compensation of the resonantdispersion introduced by an optically dense atomic mediumcan be efficient for optical depths as high as α0� � 21 000using a 640-pixel pulse shaper (note that for these opticaldepths, the attenuation at the line centre is e−21 000). Due toits finite resolution, the pulse shaper cannot follow the rapidphase variation in the vicinity of the atomic resonance. Thislimits both the quality of temporal focusing and most of themaximum optical depth that can be compensated. We testedtwo procedures of compensation. In one case, the energyspreading out of the central peak is reduced but the spec-trum is profoundly altered near the atomic resonance. In the

2 The strong variations induced on the transmitted intensity are smoothedbecause of the finite resolution of the spectrometer (� 0.1 nm)

Applied Physics B – Lasers and Optics

other case, the alteration of the spectrum is significantly re-duced but more energy is left in the wings of the central laserpeak. Depending on the fixed goal, one can use one or theother method of compensation. The compensation methodwas demonstrated on a single atomic resonance. The temporalfocusing (compression) of ultra-short pulses is of general in-terest for selective excitation of a particular medium slice. Forinstance, it is used in nonlinear microscopy (Third HarmonicGeneration) [24]. The realization of the ideas developed inthe present paper is, in principle, feasible in the situation ofpropagation in an atmosphere containing numerous absorp-tion lines due to the presence of water vapour. In this case,the spectral response can be rather complicated and/or diffi-cult to describe analytically; an optimization algorithm couldthus be used to achieve the best compensation monitoringthe cross correlation of the scattered signal. This might beconsidered in the future and would be of high interest for re-mote sensing. Indeed, the introduction of nonlinearity in thedetection scheme makes the latter sensitive to the peak inten-sity of the pulse. The position along the propagation axis atwhich the peak intensity is maximum can be manipulated bychanging the compensation introduced by the pulse shaper.This would give access directly to the local density of gas orparticles.

Finally, the extension of this method to a nonlinear inter-action regime raises new possible investigations, since propa-gation effects also affect the intensity spectrum and/or spa-tial profile (self-phase modulation, filamentation). One canalso use a pulse shaper to tailor an ultra-short pulse to con-trol the light shift of an atomic system [25, 26]. Moreover,the use of a pulse shaper in a closed-loop scheme [9, 27]may, for example, determine the form of incident shapedpulses for which either distortion effects are minimized dur-ing propagation or spectral modifications fit a desired targetshape.

ACKNOWLEDGEMENTS We acknowledge financial supportfrom the European Union Marie Curie program (Contract No. HPRN-CT-1999-00129, COCOMO).

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