An investigation of the effects of experimental parameters on the closed-loop control of...

Spectrochimica Acta Part B 58(2003) 1097–1108

0584-8547/03/$ - see front matter� 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0584-8547Ž03.00073-9

An investigation of the effects of experimental parameters on theclosed-loop control of photoionizationydissociation processes in

acetophenone�

Paul Graham, Getahun Menkir, Robert J. Levis *1,

Chemistry Department, Wayne State University, Detroit, MI 48202, USA

Received 14 November 2002; accepted 24 March 2003

Abstract

The photodissociation channels of acetophenone(C H )–CO–(CH ), can be controlled by the use of tailored6 5 3

strong-field laser pulses together with a feedback loop incorporating an adaptive algorithm. This optimal controlstrategy is used to selectively cleave either the OC–CH or OC–C H bonds, monitored by detecting either mass 1053 6 5

or 77, respectively. Varying the pulse chirp and duration prior to optimization is shown to affect the dynamic rangeof control. We show that it is possible to decrease the search space by limiting the retardance range of the spatiallight modulator (SLM), or by decreasing the number of frequency elements manipulated by the SLM, and stillachieve a certain degree of control over acetophenone dissociation. Performing consecutive experiments with identicalexperimental parameters and search criteria reveals that the learning algorithm may find solutions that have the samedegree of control(various local solutions), with either similar SLM retardances or markedly different retardances.Comparison of the dynamic range of control between single-parameter optimizations(pulse energy and duration)with the tailored electric field profiles generated by the adaptive algorithm reveals an enhancement in the control ofreaction product distributions in the latter scheme.� 2003 Elsevier Science B.V. All rights reserved.

Keywords: Spatial light modulator(SLM); Acetophenone; Photoionization

� This paper was presented at the International Conferenceon Laser Probing(LAP-2002), held in Leuven, Flanders,Belgium, July 2002, and is published in the Special Issue ofSpectrochimica Acta Part B, dedicated to that conference.

*Corresponding author. Tel.:q1-215-204-5241; fax:q1-215-204-6179.

E-mail address: [email protected](R.J. Levis).Present address: Department of Chemistry, Temple Univer-1

sity, Philadelphia, PA 19021, USA.

1. Introduction

During a chemical reaction there is typically avariety of competing product channels. One of theprinciple objectives in the chemistry concernsguiding the outcome of a given chemical reactiontowards the desired product, at the expense ofother unwanted products. The possibility of con-trolling the dynamical evolution of a reaction usingoptical fields has been the subject of intensetheoretical and experimental studiesw1–7x. Intra-

1098 P. Graham et al. / Spectrochimica Acta Part B 58 (2003) 1097–1108

molecular processes such as fast relaxation, non-adiabatic effects, intramolecular vibrationalredistribution(IVR), and an imperfect knowledgeof internuclear potentials when molecules areexposed to intense laser fields, complicate theoptical control of all but the simplest of chemicalreactions. These effects make calculation of therequired pulse shapes to attain a desired targetstate essentially impossible. To surmount the chal-lenge posed by our limited knowledge of themolecular Hamiltonian, a closed-loop controlmethod was proposed by Rabitz and coworkersw7,8x. In many closed-loop optical control experi-ments, a time dependent laser field,´(t), maysteer a reaction towards a specific product. Here,´(t) is optimized via a feedback control loop usingthe resulting product distribution(from the ´(t)-molecule interaction) to iteratively converge upona suitable laser field solution.

Advances in laser pulse shaping techniques haveallowed the development of powerful new toolsfor altering the product distribution among possiblepathways using laser excitation in the strong fieldregimew9x. Several groups have employed geneticalgorithms together with feedback control loopsfor teaching lasers to control quantum systems.Those performed to date employ adaptive closed-loop control of a laser pulse shaper to optimize adesired process. Many diverse investigations havebeen demonstratedw9–22x. A thorough account ofthe historical and technical aspects of reactioncontrol is given in the review by Levis and Rabitzw23x.

Optimization of´(t) for chemical control oftenemploys a genetic algorithm(GA) in the feedbackloop to create the trial laser fields. In the geneticalgorithm approach, the success of the few bestpulse shapes in previous generations is largelyresponsible for the generation of new individualpulse shapes that constitute the subsequent gener-ation produced via the operations of mutation andcrossover. The algorithm must be able to handle amulti-dimensional search-space and be robust toexperimental noisew8x. Furthermore, no a prioriknowledge of the molecular Hamiltonian is avail-able for input. The GA technique is well-suited toeach of these requirementsw12,24x. Adaptive fspulse shaping has been achieved with differing

types of search-space parameterizations that theGA may draw upon to the structure of its searchfor a target solutionw16x. Time and frequencyparameterization is a logical progression from thepump-probe or frequency phase-based interpreta-tion of control used by Brumer and Shapirow1,25xand the time-evolution-based interpretation by Tan-nor, Kosloff and Rice w3,4x. Parameterizationmakes convergence more probable and achievableat a faster rate since the search space is reducedin size and there is a ‘pattern’ imposed upon thesearch that the GA can utilize to find a solutionquickly and efficiently.

Control of photo-induced processes requiresoptimization of the overlap of an evolving molec-ular state±c(t)) with that of the desired targetstate (at tsq` in this case). The use of theclosed-loop paradigm is emerging as a promisingmeans of achieving such quantum control for thecase of chemical reactions. The first closed-loopcontrol experiment was realized by Wilson et al.w20x in the optimization of electronic populationtransfer in the dye molecule IR125. Closed-loopcontrol on chemical systems was then carried outby Bucksbaum et al.w12x to optimize secondharmonic generation in a non-linear medium, dis-sociative ionization in gaseous Na , stimulated2

Raman emission and self-phase modulation inliquid CCl and other applications such as coherent4

control of pulsed X-ray beamsw26,27x, shapedterahertz pulsesw28x and control of vibrationalmotion at room temperaturew29x. The applicationof closed-loop experiments for controlling selec-tive bond dissociation was first realized in the gas-phase by Gerber and coworkersw22,30,31x for anorganometallic molecule.

Strong-field closed-loop methods were recentlyemployed to control the outcome of unimoleculardecomposition of a series of ketonesw9x. Forelectronic processes, strong fields are characterizedby laser intensities greater than;10 Wycm . At12 2

intensities of 10 Wycm , the amplitude of the13–14 2

electric field of the radiation approaches the bind-ing energy of the valence electrons in molecules(approx. 10 eV). Such strong fields are known tomarkedly alter the molecular eigenstatesw32,33x.The field free discrete energy states are both Stark-shifted and broadened by the application of the

1099P. Graham et al. / Spectrochimica Acta Part B 58 (2003) 1097–1108

electric field of the laser to produce quasi-contin-uum of energy states, which can be accessed byvarious multiphoton processes.

The control of photochemical reactions inducedby the strong field femtosecond laser pulses inclosed-loop optimization experiments is describedherein. The extent of quantum control exerted overselected gas-phase dissociationyionization reac-tions in the molecule acetephenone is investigated.The effects of experimental parameters such aspulse chirp and phase retardance range on theoptimization efficiency are first presented. Theeffect of reducing the parameter-space on thedegree of control is then presented. The reductionof parameter space is accomplished through para-meterizations of the spatial light modulator(SLM)pixel groupings. Comparisons to one-parametercontrol schemes(laser pulse energy andyor dura-tion) are then presented. The optimal pulse dura-tion for control of the molecular reactions is theninvestigated. Finally, we address the propensity ofthe GA to converge on thesame minima in thesearch space in the strong-field regime, givenidentical experimental conditions(apart from theinitial randomly generated SLM optical masks)and target criteria.

2. Experimental

The laser system is based on an Ar-ion pumpedfs oscillator producing an 86 MHz train of 20 fspulses. The pulses are then regeneratively ampli-fied from nJ to mJ level at a repetition rate of 10Hz after being stretched to 10 ps duration. Theamplified pulses are then compressed using agrating pair to approximately 60 fs duration.

In the closed-loop experiments repeated here, acomputer-controlled pulse shaper(zero-lengthstretcher and SLM) generates the time-dependentelectric fields and an experimental signal, resultingfrom the interaction of the field with the systemwas used as a feedback signal to a learningalgorithm. The initial guess was improved itera-tively until an optimal solution was reached. Aliquid-crystal spatial light modulator(LC-SLM)placed in the Fourier plane of either the pulsestretcher or a zero-length stretcher, producesshaped pulses via the pixel retardances set by the

genetic algorithm. A similar degree of control wasalso achieved with the pulse shaper placed eitherbefore or after the regenerative amplifier. The SLMconsists of two arrays of 128 liquid crystal pixelssandwiched together. Modulation of pixel retard-ance allows the phase and amplitude modulationof the spectral components of the frequency-dis-persed input pulsew34x. This modulation isachieved by varying the voltage applied to thepixels (optical masks), which in turn determinesthe relative phase retardance and transmission ofeach spectral component through the SLM. Thevoltages were assigned by the genetic algorithm(GA). The initial optical masks, or individuals, inthe first generation were arbitrary in structure.From this starting point, the GA attempts toconverge upon the global ideal solution by per-forming the operations of crossover and mutationon copies of the individuals surviving proportionalselection from the previous generation. The muta-tion rates and crossover can be controlled by theexperimenter. In addition, two exact copies of thebest gene were carried over to the next generation(cloning). Each voltage array or ‘optical mask’constitutes a genome(or individual) and eachgeneration is composed of 40 individuals. The GAwas the controlling element within a feedback loopand the experimental output(mass spectrum)served as the feedback signal. The reaction prod-ucts of the molecular interaction with the shapedpulse were averaged over 10 shots. The resultingaveraged spectrum was evaluated for fitness—thatis, how well the laser field profile,́(t), drives thesystem to the target by the criteria of mass peakintensity or the ratio of mass peak intensities.

To reduce the search space the number ofindependent pixels in each array may be reducedfrom 128 to some lower value. This is accom-plished in this investigation by tying a number ofpixels together so that the same voltage is appliedwithin that sub-group. In these experiments 8pixels were tied together to give a total of 16 freevariables.

The shaped laser pulses are focused to highintensities in the interaction region between theextraction plates of the mass spectrometer. Theacetophenone vapor was delivered into the inter-action region at 300 K using a variable leak valve


Fig. 1. Strong-field, transform-limited, TOF mass spectra ofacetophenone taken with 800 nm, 60 fs duration pulses and anintensity of;10 Wycm .13 2

Scheme 1. Two selected fragmentation pathways for acete-phenone, irradiated with tailored strong-field laser pulses.

to maintain a pressure of 10 Torr inside they5

vacuum chamber. The mass spectrometric analysiswas performed using a dual slope linear time-of-flight (TOF) mass spectrometer as described else-where w35x. The laser beam was focused by a 20cm-focusing lens in a high-vacuum chamber(10 Torr sample pressure), between 2 extractiony6

plates that are 1 cm apart. The diameter of theaperture in the extraction plates is approximately0.5 mm.

3. Results and discussion

3.1. Mass spectra of acetophenone

The transform-limited mass spectrum of aceto-phenone taken at;10 Wycm , 800 nm and 6013 2

fs pulse duration is shown in Fig. 1. There are aseries of peaks present atmyz values of 15, 28,43, 77 and 105 corresponding to the CH , CO,3

CH CO, C H and C H CO ions, respectively.3 6 5 6 5

There is also a prominent peak at mass 92, corre-sponding to toluene ion( ). The featuresqC H CH6 5 3

corresponding to the unimolecular dissociationpathways in the strong field excitation of aceto-phenone are summarized in Scheme 1 showing:(a) the cleavage of the H C–CO(C H ) bond;(b)3 6 5

the cleavage of C H –CO(CH ) bond. These, and6 5 3

other energetically allowed reaction channels occurin the ensemble of molecules found at the focusof the laser. The mass peak corresponding to COfor example, must arise from a ladder climbingprocess during the transform-limited pulse thatleads to a subsequent dissociative ionization. Pre-vious investigations in the weak-field limit revealthat the photodissociation of ketones proceeds viaa stepwise primary and secondary cleavage mech-anism w36x with each step requiring several hun-dred fs to picoseconds for dissociation to occur,respectively. Here we focus on dissociation reac-tions leading to cleavage of a single C–C bond asoutlined in Scheme 1.

3.2. Investigating the effects of experimental par-ameters on optimal reaction control

The phase and amplitude-space that the GA hasaccessed for optimization of́(t) is astronomicallylarge when all possible pixels are manipulated overtheir full voltage range. This immediately suggeststhe question as to whether the search space maybe limited without affecting the effectiveness ofthe GA. This may affect not only the possibilityof convergence upon viable solutions for´(t), butalso on therate of convergence to these solutions.This question is addressed by varying the range ofthe applied SLM pixel voltage(and hence theamplitudeyphase modulations) as well as thechanging the ‘resolution’ of pulse shaping(bytying a certain number of adjacent pixels together).In these experiments, there are a maximum of 256independent variable retardance parameters corre-sponding to the number of liquid-crystal pixels(2=128) in the pulse shaper. Each pixel retardancemay be varied by controlling the applied voltagebetween 0 and 10 V with 12 bits of precision(giving 4096 levels). This provides a search space


Fig. 2. Calibration curve of the SLM, showing transmittancevs pixel driving voltage. The curve was obtained by tying onewhole pixel array to maximum voltage, while measuring thetransmission signal by a photodiode when the other pixel arrayvoltage was varied.

Fig. 3. An investigation of the effects of maximum phaseretardance as set by the maximum SLM pixel voltage withcomparison to Fig. 4. The figure above shows the ion yield ofmyzs77 and 105 separately, as well as the relative mass 105over 77 ion-yield, as a function of generation for a maximumSLM pixel voltage of 5.5 V.

of considerable size. For instance, there are256 combinations if there is one degree of phase360

control over 2p radians of retardance. The searchspace of the genetic algorithm has been limited inprevious investigations to find an optimal laserfield w9,16x. Using different characterizations inboth time and frequency-domain, Hornung et al.w16x investigated the effects of restricting experi-mental parameter space upon the optimizationprocess. Different parameterizations represent cer-tain control mechanisms and through restrictionsin these parameterizations, a faster convergencetowards a solution was found to occur. Further-more, a solution could be found in very complexsystems. Finally, a reduction in the number ofindividual parameters facilitates elucidation of thedominant control mechanisms.

Both the retardance range and the spatial reso-lution of the pulse shaper were restricted to max-imize the rate of finding a solution. A commonretardance value was assigned to a cluster of eightadjacent pixels to restrict the spatial resolution ofthe pulse shaper. This decreases the number ofoptimization variables from 256 to 16. Anotherparameter that affects the size of the search-spaceis the voltage range for the pixel(specifying theLC pixel retardance, which in turn is a measure ofthe phaseyamplitude modulation). A calibrationcurve of transmitted signal vs. pixel voltage isshown in Fig. 2. This curve was measured for

array 1 by assigning 10 V to each pixel in array2, while sweeping the voltage assigned to all ofthe pixels in array 1. The transmitted light throughthe SLM was measured as a function of thevariable voltage applied to array 1 using a photo-diode. The fixed and variable arrays were thenswitched to determine the curve for array 2. Thecalibration curves for the two pixel arrays aresimilar. This figure shows that when the voltageis between 0 and 5.5 V, the maximum amount ofvariation induced in the retardance and hencephaseyamplitude modulation is produced. Thismay be limited further to a range of 1.2–2.5 V,without much loss of generality, i.e. this rangecovers a phase shift of 2p.

The GA may be programmed to limit the max-imum driving voltage applied to the pixels of theSLM. The appropriate range of pixel retardancefor consistently and optimally controlling the pho-todissociation product ratio was considered withthe goal to limit the search-space while maintain-ing the GA’s effectiveness. A linear decrease inthe retardance range exponentially decreases thesearch space. Thus, the objective is to determine


Fig. 4. An investigation of the effects of maximum phaseretardance as set by the maximum SLM pixel voltage withcomparison to Fig. 3. The figure above shows the ion yield ofmyzs77 and 105 separately, as well as the relative mass 105over 77 ion-yield, as a function of generation for a maximumSLM pixel voltage of 1.9 V.

Fig. 5. An investigation of the duration and chirp of theunshaped seed pulse to the SLM on the ability to control withcomparison to Figs. 6 and 7. The above figure shows the rel-ative ion-yield ofmyzs105 to 77 at a maximum pixel voltageof 5.5 V, as a function of generation, at a compressor gratingposition corresponding to 60 fs pulse duration.

the limit of control as a function of the retardancerange. The manipulation of the branching ratio ofC H COyC H (phenyl cleavage to methyl cleav-6 5 6 5

age reaction in acetophenone) is the object ofcontrol. Fig. 3 displays a typical plot of this ratioas a function of generation when the maximumpixel voltage is specified to be 5.5 V(correspond-ing to a retardance range of nearly 4p). Thedynamic range in this experiment is of approxi-mately 2.5. When the maximum voltage is set tobe 1.9 V(corresponding to a phase-shift range of0–p) the dynamic range is limited to 1.3. Thegreater retardance range gives rise to a greaterdegree of control. This may be anticipated sincethe algorithm corresponding to Fig. 4 has an over-restricted search-space(ranging over less than 2pof relative phase retardance from Fig. 2) for amaximum pixel voltage of 1.9 V. Thus, viableregions of solution-space cannot be found. Analy-sis of the SLM calibration curves reveals that apixel voltage range between 1.2 and 2.5 V presentsa unique search-space of 0–2p phase shift.

Another example of limiting the search-space isto restrict the number of spectral components used

to construct the time-dependent electric field. Thismay be done in practice by reducing the bandwidthin the input laser pulse by spatial filtering of thebeam in the Fourier plane as performed previouslyw9x. Alternatively, corresponding SLM pixel group-ings may be held at constant voltage so that, whilethe frequency components still contribute to themolecular dynamics, the GA is unable to modulatetheir retardance. This will not affect the bandwidth,but will introduce an unshaped component into thepulse. Both pulse shaping and bandwidth restric-tion increase pulse duration and possibly affect therange of control of the dynamics. For instance, ithas been proposed that one reason for the limiteddynamic range in the control of nuclear dynamicsis that the same pulse is used to affect control aswell as ionizationw23x. A pulse for control of thenuclear wavepacket is expected to be of durationcomparable to the timescale of nuclear motion,approximately a few ps, whereas the ideal pulsefor ionization is one having the shortest possibleduration, approximately tens of femtoseconds. Thisleads to the question of the most appropriate pulseduration for these control experiments.


Fig. 6. An investigation of the duration and chirp of theunshaped seed pulse to the SLM on the ability to control withcomparison to Figs. 5 and 7. The above figure shows the rel-ative ion-yield ofmyzs105 to 77 at a maximum pixel voltageof 5.5 V, as a function of generation, at a compressor gratingposition corresponding to 750 fs pulse duration with positivechirp.

Fig. 7. An investigation of the duration and chirp of theunshaped seed pulse to the SLM on the ability to control withcomparison to Figs. 5 and 6. The above figure shows the rel-ative ion-yield ofmyzs105 to 77 at a maximum pixel voltageof 5.5 V, as a function of generation, at a compressor gratingposition corresponding to 870 fs pulse duration with negativechirp.

To systematically determine the effect of pulseduration on the extent of control, the separation inthe stretcher gratings can be altered to modulatepulse duration. Figs. 5–7 show experiments opti-mizing the branching ratio of C H COyC H as a6 5 6 5

function of generation for three different gratingseparations. Fig. 5 displays the relative ion yieldsas a function of generation beginning with agrating separation producing the shortest possiblepulse duration, 60 fs. The modulation for mass 77is very small, whereas the increase in mass 105 issignificant. The GA thus achieves the target ofincreasing the ratio by increasing the 105 ionsrelative to 77. The optimization is approximately1.6:1. Fig. 6 displays the branching ratio of mass105 to 77 beginning with a compressor gratingseparation producing a 750 fs pulse duration witha positive chirp. As seen in Fig. 5, the mass 77peak intensity is approximately constant, whereasmass 105 is enhanced by the GA, resulting in arelative increase of approximately 1.6:1. This opti-mum relative yield is reached more rapidly than

in the case where the compressor grating separa-tion corresponds to the transform-limited pulse,being obtained at generation 10 vs. 25. Fig. 7gives the ion yields for mass 105 and 77 as afunction of generation for a pulse duration of 870fs with a negative chirp. Again, the mass 77 peakyield is roughly constant, but the mass 105 yieldfluctuates much more and only increases by amaximum of 1.3:1 by generation 20. In this exper-iment, for positively chirped laser pulses(redfrequencies preceding blue frequencies), the degreeof control (at least for this reaction) is greater andconvergence onto the target state is achieved morerapidly. As with all experiments of this type, theexperiment had to be repeated several times toensure that the GA was not locking onto a localminima. This is compared to the trivial controlexperiment of the next section, where the relativeyields were nearly constant for all grating separa-tions when no pulse shaping was performed usingthe SLM.


Fig. 8. Ratio of ion yield of mass 105 to 77 as a function ofcompressor grating separation, which determines pulse chirpand the sign of the chirp(0 corresponds to a transform-limitedpulse).

3.3. Trivial control of molecular fragmentation

There is always the question of whether simplercontrol schemes, such as controlling pulse durationand energy, would suffice to achieve the soughtafter reaction outcome. Manipulation of the laserpulse duration or energy, such that the phase andyor amplitude of the spectral components are affect-ed equally, is defined here as ‘trivial’ control. Thecoherent control technique with feedback and pulseshaping is compared with the trivial approach toascertain whether complex time-dependent laserfields are really necessary. The branching ratio ofC H COyC H was measured as a function of6 5 6 5

laser pulse duration or ‘chirp’, as controlled by thecompressor grating separation. The amount andsign of the pulse chirp may potentially controlpopulation dynamics, as was tested in the dissoci-ative ionization in Na by Bucksbaum et al.w12x.2

In that experiment no evidence was observed forchirp control. However, Bardeen et al.w37xobserved that population transfer depend on thesign of the chirp. This is also the conclusion ofBrixner et al. w10x, where the phase structure ofthe pulse was shown to be important in the laserymolecule interaction. This was attributed to theinitial wavepacket excitation being initiated byfrequencies that occur early in the pulse, followedby wavepacket evolution and subsequent interac-

tions with spectral components that occur later inthe pulse.

Fig. 8 shows the ratio of the mass peaks 105 to77 as a function of compressor grating separation.The grating separation both specifies the laserpulse duration and the temporal order of thefrequency components of the pulse. The pulsedurations shown in Fig. 8 are calculated from thecompressor grating separations. The pulse durationranges from 730 fs for the smallest grating sepa-ration corresponding to a positive chirp, to 874 fsat the largest grating separation corresponding toa negative chirp. Brixner et al.w10x proposed thatstretching the pulse duration via a quadratic chirpcontrolled the branching ratio of .q qFe(CO) yFe5

However, observation of Fig. 8 shows that in thepresent case of dissociative ionization in acete-phenone, neither the sign of the chirp nor the pulseduration is important for obtaining control.

Varying the laser pulse energy was also per-formed as part of the series of reference experi-ments. The energy was attenuated usingcover-slides of known transmission efficiency andthe pulse energy was continuously monitored. Ananalysis of pulse duration revealed no significantincrease upon insertion of the 200mm thick slides.Attenuating the energy merely reduces the frag-ment ion yield, while leaving the relative ion ratioconstant. This is in contrast to the system studiedby Brixner et al.w10x where control of the branch-ing ratio of could be achieved byq qFe(CO) yFe5

attenuating the laser pulse energy. This was attrib-uted to Rabi cycling or coupling between dressedstates within the molecule, although a ladder-switching or some other intensity-dependent mech-anism may also be responsible.

Trivial control schemes do not generally allowfor exploitation of molecular properties such aselectronic, vibronic and dynamical effects of thelaserymolecule interaction. Such trivial control wasalso found not to allow control in liquid-phaseexperiments performed by Gerber and co-workersw38x. Monitoring the optimization of a target ratioof liquid-phase molecular excitations and simulta-neous optimization of SHG signals, it was dem-onstrated that single-parameter search-space doesnot allow control. Furthermore, a greater degree ofcontrol was obtained when all 128 pixels were


Fig. 9. (a–d) Four separate GA trials with associated optical masks(inset) for equal experimental conditions. Fig. 9b and Fig. 9cshow similar characteristics, which are reflected in the shape of the optical masks.

treated independently, rather than tying them intosub-groups. This suggests that coherent controlgenerally requires a more complex laser pulseprofile than a bandwidth-limited Gaussian and thatthe multi-dimensional search space topology isgenerally complex.

3.4. Convergence to the same minima

Inspection of Figs. 5–7 suggests that optimiza-tion occurs by manipulation of the mass 105 peakintensity, while maintaining constant intensity in

mass 77. This raises an interesting question: doother solutions exist to achieve similar relative ionyield optimizations? Previously, in results withacetophenonew9x, the solutions found by the GAmaintained a constant mass 105 peak intensity,while altering the mass 77 peak. This suggests afamily of possible target solutions, when the targetcriterion simply corresponds to an increase in theratio of peak intensities. We note that for experi-ments performed inw9x, the laser pulse made adouble pass through the SLM, which was locatedin the Fourier plane of the pulse stretcher and the


bandwidth of the pulse was smaller than at present.This affects the pulse duration of the solutionsavailable to the GA. In the case of the experimentsperformed inw9x the minimum pulse duration ofthe shaped pulse was 300 fs. Perhaps, counter-intuitively, smaller excitation bandwidths allow agreater degree of control to be achieved due to thelonger pulse duration.

To determine whether the GA converges to thesame local minimum every time, or is able to finddifferent local minima, the GA was run for severalconsecutive experiments using identical initial con-ditions and target specifications. Fig. 9a–d showthe branching ratio of C H COyC H and the6 5 6 5

corresponding ‘best-fit’ SLM optical masks for aseries of four consecutive runs with identicalfitness criteria. Analyzing the optimization of the105y77 ratio in Fig. 9a and b, the GA optimizedthe ratio by a factor of 2 and 3, respectively, andthe optical masks generated by the GA possess asimilar shape. In Fig. 9a, mass 77 and 105 wereboth increased and the ion ratio steadily increasedtwo-fold by about the 30th generation. In Fig. 9b,the ratio increased to 3:1 by the 30th generation.Two other runs are shown in Fig. 9c and d, bothof which are visibly different from each other(andfrom the first two runs). In Fig. 9c, mass 77 againincreases by about a factor of 2, whilst mass 105increases very sharply at generation 21 to about afactor of 3.5, whereupon the mass 105 peak expe-riences large modulations between 4.3 and 3 timesits original value. In Fig. 9d, mass 105 and mass77 track each other closely, both increasing anddecreasing in similar fashion, until about genera-tion 65. After generation 65, mass 105 increasessteadily while mass 77 fluctuates and the relativeion yield also increases until the 75th generation.The optimization of the branching ratio increasesby 2 and 2.5 in Fig. 9c and d, respectively. Thisis comparable to the results obtained in Fig. 9aand b suggesting a series of different local minimaall giving rise to similar optimizations.

It follows that from this set of experiments thereare many reaction pathways that the algorithmmay follow in order to achieve a specific goal andif left long enough, will likely converge upon onesuch solution. Furthermore, the final masks thatare obtained are not always similar, i.e. the algo-

rithm does not converge upon a single localminimum.

4. Conclusions

The conclusions arrived at from these sets ofexperiments are that the strong-field interaction ofacetophenone may be driven towards a desiredend-product by means of complex time-dependentelectric fields,́ (t). This is achieved through recentpulse-shaping techniques and by allowing the mol-ecule to provide feedback in real-time to theadaptive GA learning-loop. This allows for aconvergence upon a solution of the electric fieldby an iterative process. Experiments that affectsimple control, i.e. variation of laser chirp or pulseenergy do not significantly alter the moleculardynamics in the experiments detailed here.

The effect of varying experimental parametersthat have an impact on the size of the search spacethat the GA can access is important in the controlof molecular dynamics. Restricting the searchspace by tying a number of SLM pixels togetherand by limiting the maximum phaseyamplitudemodulation that may be applied by each pixel set,is shown to be useful for the GA to converge upona target solution. The pulse durationybandwidth ofthe pulse may also affect the reaction in optimalcontrol experiments as it determines how thereaction evolves and also the pulse duration shouldbe comparable to the reaction time-scale. Finally,the GA does not always converge upon the samesolution field to achieve a desired product distri-bution, but is able to converge upon any of aseries of possible solutions for identical searchcriteria.

Acknowledgments

The authors acknowledge the Office of NavalResearch, the DoD MURI program as managed bythe Army Research Office, the National ScienceFoundation and the Sloan and Dreyfus Foundationsfor the support of this research.

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