Feedback Optimization Strategy for Rotational Alignment Echo Spectroscopy

The recent discovery of rotational echoes has contributed to extend the applications of field‐free molecular alignment to short time scales, namely to a temporal region preceding the first alignment revival of the molecule. Most echo measurements require adjusting the position of the echo over time, which in the case of molecular alignment echo unavoidably leads to a change in its amplitude, unless the intensity of the pulse sequence triggering the echo is properly modified. Herein, it is proposed to avoid this drawback by using an optical pulse shaper steered by a learning algorithm. It is demonstrated that a temporal shaping, whose characteristics are optimized to maintain a constant echo amplitude regardless of its creation time, can be generated by controlling the spectral phase and amplitude of the laser pulse which are then used in an open‐loop control system. As a proof of principle, the optimization strategy is applied to the observation of short‐term dissipative dynamics of laser‐aligned molecules exposed to collisions and to the measurement of associated decoherence and population decay time constants.


Introduction
4][5] The attention paid to this new phenomenon is mainly motivated by one of its unique properties, namely, the ability to generate field-free ensembles of aligned molecules at a predetermined time, independently of the molecular system.To understand the interest of rotational echoes, recall that in "standard" impulsive laser-induced molecular alignment (for a review on molecular alignment and orientation see, e.g., refs.[6-10]), the rotation of the molecules is triggered by a single pump pulse through impulsive nonresonant Raman excitations.This results in an initial alignment of the molecules vanishing shortly after the pulse interaction due to dispersion of the angular velocities associated with excited rotational states of different energy.Thanks to the quantization of the rotational motion and to the initiating pulse of a much shorter duration than the rotational period of the molecules, the initial alignment is periodically recovered leading to the so-called alignment revivals. [11,12]Note that this would continue indefinitely if dissipation [13] and centrifugal distortion [14] were not taken into account.At very low temperatures, molecular alignment experiments performed with, for instance, quantum state selection, [15] will show a dependence of the recurrence of the revivals on the characteristics of the laser pulse and the energy difference between the excited rotational states. [16]or these specific conditions, it has been shown that rich "continuous" quantum dynamics can be revealed by observing the angular distribution of the aligned molecules. [17]owever, for molecular samples at room or moderate temperature, such as those relevant to the type of applications addressed in the present work and many others, the time interval between revivals is determined by the moment of inertia of the molecule, [8,16] the smaller the second the larger the first.The fact that the timing of the revivals is tightly bound to the molecule sets a limitation for applications (see later) in which the alignment occurrence must be adjusted independently of the system.On the contrary, rotational echoes, as classical effects, [5] can potentially be produced at any time t E and therefore do not suffer from this drawback.Similarly to spin and photon echoes, rotational echoes are produced by applying to the system two pulses, hereafter referred to as P 1 and P 2 , time delayed by τ 12 .We have shown that it leads to the creation of a main alignment echo centered at time t E ¼ 2τ 12 , [1] followed by higher-order echoes, as well as fractional, [18] imaginary, and rotated echoes. [19]The adjustability of t E has been exploited for probing ultrafast collisional relaxation of linear (CO 2 ) [20] and symmetric-top molecules (C 2 H 6 ), [21] in pure and gas mixtures, for pressure ranges where rotational decoherence could not be measured using standard alignment revivals. [13,22]By observing the pressure-induced relaxation of the rotational echo during the first few picoseconds following the excitation of N 2 O molecules diluted in He, time-domain observation of nonsecular collisional transfers occurring between rotational coherences could be achieved for the first time. [4]his methodology has been recently applied to observe dissipative non-Markovian open quantum system dynamics in CO 2 and HCl molecules. [23,24]The adjustability features of the echo were also exploited to significantly improve the alignment of the acetone molecule at room temperature, enabling the third-harmonic generation from circularly polarized light otherwise not detectable through alignment revivals of this molecule. [25]ther applications of rotational echoes can be found, for instance, compensation of the centrifugal distortion of CH 3 I molecule, [14] calibration of the degree of laser-induced molecular alignment, [26] unidirectional rotation of molecular rotors, [27] and high-order harmonic generation. [28]n echo spectroscopy, it is usual to perform measurements for different time occurrences of the echoes.For instance, measuring the decay of the photon echo [29] amplitude versus the delay between the two excitation pulses provides relevant information about decoherence time for homogeneously and inhomogeneously broadened systems. [30,31]This kind of study is possible because the magnitude of a photon echo does not intrinsically depend on the delay between the two excitations.This is no longer valid for a rotational echo where its amplitude is strongly determined by t E .This effect finds its origin in both the filamentation of the phase space in classical analysis [1] and the multilevel nature of the rotational motion in quantum analysis. [3]However, it has been shown [3] that for each value of the delay τ 12 , one can find an intensity (I 2 opt ) of P 2 that maximizes the echo to the same amplitude regardless of τ 12 .In practice, the fine adjustment of I 2 is not necessarily easy to implement, especially in quantitative studies involving low signal-to-noise ratio and/or requiring long series of data acquisition.This was for instance the case in the previous studies [20,21] where, to improve the accuracy of the measurements, the variation of the delay τ 12 was limited over a small range so that I 2 opt could be considered constant overall measurements.However, note that this approach is only applicable to studies where the intrinsic variation of the echo with τ 12 is small compared to the effect to be measured.
In the present work, we investigate a process for rotational alignment echo spectroscopy which overcomes the constraint due to the intricate dependence of the rotational echo amplitude on I 2 and τ 12 .It is based on an optimization of the echo amplitude by a shaped laser pulse generated with a spatial light modulator (SLM) feedback controlled by an evolutionary algorithm.As a proof of principle, we apply the proposed method to measure the pressure-induced decay of the rotational quantum coherences and populations of aligned molecules.

Setup
The experimental setup is represented in Figure 1.The laser beam produced by a chirped-pulse amplified Ti:Sapphire laser [800 nm central wavelength, 100 fs duration full width at half maximum (FWHM), and 1 kHz repetition rate] is splitted into a pump and probe beam.The former is directed to a pulse shaper which is based on a zero-dispersion 4f line comprising a pair of 1,600 lines mm À1 gratings and 400 mm focal length cylindrical mirrors.A programmable one-dimensional dual mask liquid crystal spatial light modulator (SLM) array is located in the Fourier plane of the 4f line.A polarizer is placed at the output of the 4f line allowing independent control of the spectral phase and amplitude for each of the 320 pixels composing the SLM arrays.The tuning of the temporal delay between the probe pulse, frequency doubled in a type I BBO crystal, and the shaped pump pulse is achieved by a motorized translation stage equipped with a corner cube reflector.The probe beam is then vertically polarized at 45°with respect to the pump beam (using half waveplates and polarizers not represented in Figure 1) right before both are focused in a gas cell, the overlap between the two foci being optimized by a telescope inserted in the probe beam path.After propagation through the gas cell, the change in polarization of the probe beam is analyzed by a balanced detection [32] providing a signal proportional to the alignment factor cos 2 θ h i T À1=3, [33] where θ is the angle between the molecule and the polarization direction of the pump pulse and h i T denotes a thermal averaging over the quantum expectation value of the observable cos 2 θ accounting for the finite temperature T of the medium.

Concave
Mirror SLM

Pulse Shaping
Figure 2a provides an example of alignment signal produced by the shaped pulse in low gas pressure of N 2 O molecules.The SLM has been programmed so that the pulse shaper produces as outcome two pulses P 1 (intensity I 1 ) and P 2 (intensity I 2 ) time shifted by τ 12 .As shown in Figure 2a, each of them is responsible for initial alignment peaks denoted IAP1 and IAP2, but the combination of both generates the main echo E1 at 2τ 12 , i.e., the echo of interest in the present work, and a second-order echo E2 at 3τ 12 .Note the presence of an elevated baseline after the first pulse excitation corresponding to the permanent alignment [8] resulting from the large intensity of P 1 (I 1 ¼ 20 TW cm À2 ).As exemplified by Figure 2b, the shaped pulse is synthesized by applying to the right half-part of the SLM input spectrum (i.e. for ω > ω 0 , where ω 0 is the central frequency) an intensity transmission factor TðωÞ and a linear phase shift φðωÞ ¼ ωτ 12 .These two parameters determine the delay τ 12 and the field intensity ratio R ¼ I 2 I 1 ≤ 1 between the two pulses, with the pulse P 1 carrying half of the total input energy.The applied phase and transmission modulations are pixelated.Recall that for a given pixel i (1 < i < 320) of the liquid crystal SLM array, the transmission and phase are given by , where ϕ 1i and ϕ 2i are the phases induced by each pixel i of masks 1 and 2, as depicted in Figure 1, respectively.The dispersion of the pulse shaper in the Fourier plane has been calibrated to estimate the averaged value of the spectral sampling Δν=61.2GHz pix À1 .This allows to assess the major pulse distortions of the SLM related to the pixelated nature of the device.As explained in the previous studies, [34,35] the shaping of the pulse inherently results in the production of replicas with a time period Δτ rep ¼ 1=Δν ¼ 16.3 ps.In addition, the temporal field consisting of the desired pulse sequence P 1 , P 2 , and replica are modulated by a sinc time window given by sincðπΔνtÞ.The output of the pulse shaper has been characterized by recording in argon the third-order noncollinear cross-correlation Kerr signal between the shaped and probe pulse.The data reported in Figure 2c have been recorded for different phase modulations φðωÞ and same transmission factor TðωÞ ¼ 1.They clearly show the pulse P 1 and P 2 generated with a time delay τ 12 , as well as the replicas produced before P 1 at t ¼ τ 12 À Δτ rep .The pulse duration of P 1 and P 2 derived from the cross-correlation traces, % 185 fs (FWHM), is in good agreement with the calculation (170 fs) obtained from TðωÞ and φðωÞ applied to the pulse shaper and the initial spectal bandwidth of incoming pulse (11 nm FWHM).

Optimization
The echo optimization is obtained by an evolutionary algorithm [36] programmed to find the best value of T and φ maximizing the peak amplitude A þ (see Figure 2a) of the echo measured by the probe pulse at time t pr % t E ¼ 2τ 12 .The optimization procedure uses a low-dimensional parameterization of the search space.The delay τ 12 and transmission T applied by the pulse shaper on half part of the spectrum to produce P 1 and P 2 build the genes of the evolutionary algorithm.Note that the total energy sent to the pulse shaper is chosen so that the half part used to generate P 2 is far above the value required to optimize the echo signal for all delays τ 12 .The two parameters τ 12 and T are iteratively optimized by the learning algorithm for maximizing the echo leading to a constant echo amplitude regardless of its creation time.The algorithm starts from a population of 30 individuals composed of randomly chosen genes, bounded within an adequate range.Each individual is experimentally tested by measuring the echo signal by means of a probe pulse timed to a given (fixed) temporal delay t pr with respect to P 1 .The fitness A þ is then used as a feedback signal for the adaptive optimization that uses the concepts of biological evolution.As shown in Figure 2d, convergence is usually achieved after 10 generations.Note that the pump-probe signal, as presented in Figure 2a, was recorded after optimization of the echo, i.e., for the optimal intensity I 2 opt of P 2 .
It should be emphasized that the optimization procedure described above is not compromised by the presence of replica and the sinc time window effect described in the previous section.The first reason is that the replica is much weaker than P 1 and therefore the first does not significantly affect the alignment induced by the second.The second reason is that the effect of the replica on the echo signal is most noticeable when the echo induced by the replica and P 1 coincides with P 2 , which in our case only occurs for τ 12 ¼ 8 ps.As will be shown in the next section, this does not noticeably affect the optimization of the echo signal.

Validation of the Method
To demonstrate the benefit of our approach, the magnitude of the echo has been recorded for τ 12 covering a broad temporal domain of 6.5 ps.As for the record of Figure 2a, the N 2 O pressure in the gas cell was reduced to 0.10 bar to avoid any noticeable collision-induced change in echo amplitude while varying τ 12 .For each pump-probe delay t pr , the outcome of the shaper was optimized to produce a maximum probe signal, prohibiting the system to overlap the initial alignment peak of P 2 (shown in Figure 2a) with the probe pulse, which occurs when t pr % τ 12 .The fully automatized record depicted in Figure 3a with red circles was obtained for a peak intensity I 1 of P 1 estimated around 20 TW cm À2 .Except the three features labeled A, B, and C (discussed at the end of this section), the variation of the echo amplitude is less than % 10% over the whole temporal range corresponding to a tuning of the echo between 2 and 15 ps.This value is to be compared to an amplitude variation of % 300% obtained when the signal is measured for a fixed intensity of P 2 maximizing the echo for τ 12 ¼ 5 ps.The optimized intensities I 2 opt versus τ 12 retrieved from the algorithm and shown in Figure 3b illustrate the strong intricate dependence of the echo with respect to these parameters.These intensities have been estimated taking into account the diffraction effects and pixellation of the SLM addressed in Section 2.2.As also shown in Figure 3a with the data depicted with blue crosses, the robustness of the method has been successfully tested in a day-today operation by programming the pulse shaper with the outcome previously optimized for each delay τ 12 without running again the evolutionary algorithm.For further validation of the results, the alignment signal was calculated by solving the Liouville von Neumann equation for different values of τ 12 and I 2 using Fourier-transform-limited (FTL) pulse P 1 and P 2 of duration equivalent to that produced by the pulse shaper (% 185 fs).As shown in Figure 3a, the maximal peak amplitude of the echo as a function of τ 12 is well reproduced by these calculations.The same agreement is found regarding the optimal intensity of P 2 presented in Figure 3b.Note that similar calculations based on the amplitude and phase modulation achieved by pulse shaping versus the delay τ 12 between the two pulses P 1 and P 2 .The experimental data (red circles), obtained by optimizing the phase ϕ 1 and ϕ 2 with the evolutionary algorithm, are compared to numerical simulations (green line) based on FTL pulses.The data obtained by programming the shaper to apply the phases learned from the previous optimization process are depicted with blue crosses.The three features marked with arrows correspond to the overlap of the main echo with the first (A), second (B), and third revival (C) of the imaginary echo (see text), respectively.b) Optimized intensity I 2 opt of P 2 (red circles) produced by the pulse shaper for different τ 12 compared to simulations (green line).c) Overlap region between the main echo and the first revival of the imaginary echo: simulated (green line) and measured signal (red line and circles).
in the experiments lead to the same agreement confirming the weak role played by the replica.
Finally, we would like to address the fast signal modulations observed in Figure 3a (labeled A, B, and C) that are also reproduced by the numerics, as shown for instance in the highresolution measurements of Figure 3c.These features result from the overlapping of the main echo with imaginary echoes.The latter phenomenon refers to an echo signal that would be observed at negative time τ 12 before P 1 if, after applying P 2 , one were able to reverse the course of time. [19]To better understand the following, calculated alignments features (initial alignment peaks, alignment revivals, main, higher-order, imaginary echoes, and their respective revivals) are depicted as a function of time in Figure 4 for fixed P 1 and P 2 intensity.As described in the study of Lin et al., [19] thanks to the quantization of the rotational energy, the imaginary echo manifests itself at positive instants by its revivals which occur at times T r 2 À nτ 12 , where T r 2 (% 20.22 ps) corresponds to the instant of first revival of N 2 O and n refers to n th revival of the imaginary echo.As τ 12 increases, imaginary echoes (IER1 and IER2) propagate backward, causing the main echo (E1) to overlap with the first (n ¼ 1) and second (n ¼ 2) revival of the imaginary echo for 6 % 6.63 ps (A) and T r 8 % 4.97 ps (B), respectively.The resulting interferences are responsible for the signal oscillations observed in Figure 3a.Note that the third revival (n ¼ 3) of the imaginary echo IER3 (C) does not appear in Figure 4 for the reason that the fixed intensity of P 2 used for calculations not correspond to the optimal one for the delay leading to the overlap at τ 12 ¼ T r 10 % 3.98 ps.

Probing Collisional Dissipation
][39][40][41] When dissipation is induced by collisions, it has been shown that the rate of rotational decoherence (1=T 2 ) and population decay (1=T 1 ) can be inferred from the peak-to-dip amplitude of the transient alignment features (revivals [13,22] and echoes [20] ) and from the permanent component of the alignment, [13,22] respectively.The approach presented in the previous section has been applied to rotational echo spectroscopy for probing short-time collisional dissipation.
The collisional time constant τ E ð2τ 12 Þ of the alignment echo related to the lifetime of rotational coherences was determined for N 2 O in the study of Ma et al. [4] by measuring the peak-to-dip amplitude of the echo generated at time 2τ 12 as a function of the gas density d for a fixed delay τ 12 .By repeating this exercise for different τ 12 , it has been possible to observe a slowdown of the dissipation in the early stage of the system evolution for 1.5 < τ 12 < 5 ps, resulting from nonsecular collisional transfers between rotational coherences. [4]Here, we suggest to follow a different approach.By fixing the gas density d and optimizing the echo for different times t E % 2τ 12 , we propose to extract from the optimized echo an "average" value of the collisional decay time constant τ E of the echo.This requires not only to measure the maximum amplitude A þ of the echo, as shown in Figure 3a, but also its minimum value A À (see Figure 2a), which are both sensitive to coherence (echo) and population (permanent alignment) decay, to extract the peak-to-dip amplitude of the echo a quantity that is only related to rotational coherences. [13]To this aim, the algorithm driving the pulse shaper was modified so that for each couple (τ 12 opt , I 2 opt ) maximizing the echo signal A þ at time t pr , the delay τ 12 was tuned by AE400 fs around τ 12 opt by steps of 20 fs to localize and measure the minimum A À .Considering the small range of the delay, this was implemented by applying to the shaper a phase ramp, keeping P 2 intensity (I 2 opt ) constant.) and P 2 (13 TW cm À2 ) calculated for different delays τ 12 from 0.5 to 9 ps (IAP1 and IAP2, initial alignment peaks produced by pulse P 1 and P 2 , respectively; E1, main echo; E2, second-order echo; E3, third-order echo; IER1, 1st revival of the imaginary echo; IER2, 2nd revival of the imaginary echo; ARP1, 1st alignment revival induced by pulse P 1 ; ARP2, 1st alignment revival induced by pulse P 2 ).A and B: temporal overlap zones of the main echo E1 with imaginary echoes IER1 and IER2, respectively.

Closed-Loop Optimization
The experiment was performed by filling the gas cell with 10 bar of room temperature gas mixture of N 2 O(5%)-He(95%) corresponding to a N 2 O-He gas density d ¼ 9.10 amagat.The results presented in Figure 5a have been obtained by measuring the peak-to-dip amplitude of the optimized echo generated at different times t pr (red circles).The signal exhibits the exponential decay feature expected for collisional relaxation of binary mixtures.To underline the interest of the method, the experimental data were also recorded with a nonoptimized P 2 intensity, i.e., with a pulse shaper programmed to produce a bipulse with fixed I 2 value regardless of the delay τ 12 (blue squares).In that case, the variation of the echo results from the intrinsic dependence of its amplitude on τ 12 (decrease and then increase of the echo with t pr ) combined to the pressure effect (decrease of the echo with t pr ) which prohibits any direct measurement of the second effect.

Open-Loop Optimization
Another benefit of the present optimization method is the possibility to use it in a open-loop configuration.In the context of the present application, the idea was to use the parameters τ 12 opt and I 2 opt , learned from the closed-loop optimization conducted in the low-pressure conditions of Figure 3, to perform measurements in the high-pressure regime in an open-loop configuration, i.e., without implementing new optimizations for each time t pr .Such measurements are presented in Figure 5a.They have been implemented with the same gas sample as in the previous measurements.As shown, the data obtained through the open-loop method (green triangles) are similar to those obtained through the closed-loop optimization (red circles), which demonstrates that the optimization method is reliable for operating in openloop.In addition to saving time, the latter makes it possible to increase the precision of the measurements by offering the possibility to investigate experimental conditions for which the signal-to-noise ratio prohibits an accurate closed-loop optimization of the echo.This was the case in the present study, where for instance the echo recorded in Figure 5a at t pr < 3 ps and t pr > 12 ps could only be measured using the open-loop mode (or with an unoptimized I 2 intensity).The same has been applied for the pressure limit, where the open-loop approach has been used up to 20 bar, while the closed-loop optimization measurements were limited to a maximal pressure of 10 bar.To estimate the density normalized value of τ E and its uncertainty, the open-loop measurements of Figure 5a have been repeated for different densities d. Figure 5b presents the collisional decay rates τ 0 E À1 of each data set least-squared fitted by expðÀ2τ 12 =τ ' E Þ as well as the linear fit of the all sets leading to the estimated constant τ E ¼ τ ' E d ¼ 65 AE 3 ps amagat.Finally, the relaxation of the rotational populations has been obtained by measuring the amplitude  Figure 5. a) Peak-to-dip echo amplitude recorded in a high-pressure (10 bar) binary gas mixture of N 2 O and He at different probe times t pr .Three series of measurements have been performed: i) by optimizing the high-pressure echo signal using the evolutionary algorithm (red circles), ii) by using an unoptimized P 2 intensity set to I 2 ¼ 4.3 TW cm À2 for all t pr (blue squares), and iii) by programming the pulse shaper to apply the phases learned from the optimization procedure conducted in the low-pressure regime (green triangles).The exponential fit of the data measured with the open-loop procedure is depicted with a green dashed line.b) Collisional decay rates τ' E À1 of the echo measured for various densities d of the gas mixture and linear fit of the data (dashed line).
the permanent alignment versus time.Following the same methodology as in Figure 5a, we have derived an estimated collisional time constant value of the permanent component τ P ¼ 321 AE 35 ps amagat.The large error bars result from the much larger value of τ P as compared to τ E .This difference, already observed in CO 2 molecule, [22] is explained by the fact that, due to a gyroscopic effect, the orientation of the rotational angular momentum quantified by M=J, where J is the rotational quantum number and M its projection along the direction of the laser field, is much less sensitive to collisions that the rotational energy quantified by J. To improve the accuracy of τ P , measurements would need to be performed over a much longer temporal range (up to 28 ps) not achievable with the present pulse shaper.Note that to improve the accuracy of this constant, the permanent alignment has been conjointly measured by standard (single) pump-probe measurements [22] leading to τ P ¼ 344 AE 6 ps amagat.

Conclusion
We have shown that the main drawback of molecular alignment echo spectroscopy can be advantageously solved by shaping the phase and amplitude of the aligning field.The proposed method consists in using a spatial light modulator to synthesize a bipulse of fixed pulse duration but with delay and amplitude ratio controlled by two parameters optimized through a closed-loop evolutionary algorithm.We have demonstrated that such pulse shaping can generate, in a fully automated way, a tunable rotational alignment echo over 15 ps, which is more than enough for most rotational echo spectroscopy applications, with less than % 10% amplitude variation over the entire range.The optimization method has been successfully applied in a closed-and open-loop configuration to measure the collisional decoherence and population decay of aligned N 2 O molecules diluted in helium.
Compared to standard optical setup (based on spatially separated beams) for producing bipulse of adjustable delay and amplitude ratio, the present optimization method offers significant advantages.First, the two spatial profiles of the pump pulses produced by the pulse shaper perfectly overlap, which is important considering that the echo is a nonlinear process depending on the third power of the pump intensity (for moderate intensity).Second, acquisition times are greatly reduced because the setting of the delay τ 12 between the two pulses, controlled by the spatial light modulator, requires no moving optics.Third, the maximum of the echo is recorded without the need to scan the delay of the probe pulse since the system automatically adjusts τ 12 to temporally overlap the echo with the maximum of probe pulse.To quantify the benefit of the last two points, note that measurements like those presented in Figure 3a would take 5 times longer compared to a "traditional" method requiring to store 14 pump-probe traces (one for each delay τ 12 ), with a fine tuning of the intensity of the second pulse between each one.Finally, we have demonstrated that the use of the optimization procedure in a open-loop configuration saves additional time as well as gives access to a wider range of experimental parameters.For all these reasons, we believe that our results will contribute to improve rotational echo spectroscopy and thereby help to develop new applications as for instance 2D spectroscopy [42][43][44] of systems involving rovibronic resonances or probing dissipation dynamics of slowly rotating molecules like I 2 , [45] as well as molecules embedded in nanodroplets. [46,47]

Figure 2 .
Figure 2. a) Alignment features produced by pulse shaping in low-pressure N 2 O molecule.The initial alignment peaks IAP1 and IAP2 consecutive to the excitation of the molecule by the pulses P 1 and P 2 (separated by τ 12 ¼ 1.5 ps), respectively, are depicted in green, whereas the main echo E1 (of maximum and minimum value A þ and A À , respectively) and second-order echo E2 are shown in red and blue, respectively.The permanent alignment (PA) is denoted with the dashed black line.b) Spectral intensity transmission (top) and phase (bottom) generated by the combination of liquid crystal masks depicted in Figure 1.c) Cross-correlation signals (solid lines) of the SLM output recorded in argon.The results are shown for various applied phase modulations corresponding to different generated delays τ 12 between pulse P 1 (t ¼ 0) and P 2 (t > 0).The pulse replica (t < 0) and the sinc time window envelop (black dashed line) are inherent features related to the pulse shaping technique (see text).d) Evolution of the fitness function (A þ ) versus the number of iterations.The fitness of the best individuals are depicted as a function of number of iterations.

Figure 3 .
Figure3.a) Peak amplitude A þ of the alignment echo generated in N 2 O by pulse shaping versus the delay τ 12 between the two pulses P 1 and P 2 .The experimental data (red circles), obtained by optimizing the phase ϕ 1 and ϕ 2 with the evolutionary algorithm, are compared to numerical simulations (green line) based on FTL pulses.The data obtained by programming the shaper to apply the phases learned from the previous optimization process are depicted with blue crosses.The three features marked with arrows correspond to the overlap of the main echo with the first (A), second (B), and third revival (C) of the imaginary echo (see text), respectively.b) Optimized intensity I 2 opt of P 2 (red circles) produced by the pulse shaper for different τ 12 compared to simulations (green line).c) Overlap region between the main echo and the first revival of the imaginary echo: simulated (green line) and measured signal (red line and circles).

Figure 4 .
Figure 4. Echo landscape.Molecular alignment features of N2 O produced by two short laser pulses P 1 (20 TW cm À2 ) and P 2 (13 TW cm À2 ) calculated for different delays τ 12 from 0.5 to 9 ps (IAP1 and IAP2, initial alignment peaks produced by pulse P 1 and P 2 , respectively; E1, main echo; E2, second-order echo; E3, third-order echo; IER1, 1st revival of the imaginary echo; IER2, 2nd revival of the imaginary echo; ARP1, 1st alignment revival induced by pulse P 1 ; ARP2, 1st alignment revival induced by pulse P 2 ).A and B: temporal overlap zones of the main echo E1 with imaginary echoes IER1 and IER2, respectively.