Optimization of dynamic molecular alignment and orientation by a phase-shaped femtosecond pulse Arnaud Rouzee, Omair Ghafur, Arjan Gijsbertsen, Wing Kiu Siu, Steven Stolte, Marc Vrakking 9 March 8 Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Outline Definitions and goals Quantum mechanical model Evolutionary algorithm as a tool to optimize the alignment/orientation Experimental considerations: simplification Conclusion and outlook Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Definition Molecular alignment vs orientation Angular localization of one or more molecular axes along chosen directions: Alignment C=C bond axis angularly confined in one direction For orientation: Arrange molecules in a head versus tail order Alignment Orientation Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Why aligning/orienting molecules? The direction of the molecular axes plays a key role in several processes such as in strong field molecule interaction like for instance: Ionisation Dissociation pump Ep θ ϕ Absorption High order harmonic generation provides the means to study these processes without averaging over an initial angular distribution of the molecule, which is generally more suitable to the understanding of underlying physical mechanisms. Alignment or orientation as a tool to control the chemical reaction or collision process Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
State of the Art For alignment: Nonresonante interaction between a laser field and the molecular polarizability Adiabatic alignment Field-free alignment Larsen et al., JCP,, (999) -D alignment Rosca Pruna and Vrakking, PRL, 87, () -D alignment Larsen et al., PRL, 8 () 3-D alignment Lee et al., PRL, (7) 3-D alignment For orientation: Interaction of a DC electric field with the permanent dipole moment Half cycle pulse (THz pulse) Combination of two frequencies (ω,ω) Combination of a DC electric field with a nonresonant pulse Sakai and al., PRL, 9, (3) One experimental evidence of orientation using a DC electric field and a long laser pulse: orientation remains small with only a few percent of the molecules oriented towards one side. Moreover, adiabatic orientation. Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Goals Better orientation at very low temperature: Using a supersonic jet: Rotational cooling Using an hexapole state selector: Full state selection Theoretical work: Impulsive molecular orientation and alignment of a state-selected NO molecules using a DC field and a short laser pulse Optimize both the orientation/alignment using a phase shaped laser pulse and find the best experimental conditions to the observation of the optimization Experimental work: Observe for the first time the impulsive molecular orientation using an hexapole state selector See an enhancement of the impulsive orientation using the shape found by the algorithm Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Impulsive orientation/alignment EA pump Quantum mechanics: Hexapole state selector: only one initial populated state Hamiltonian: ψ i = J, Ω,ε, M Energy of the molecule H = H rot + H O + H A DC electric field Short pulse H O = µ p EO cos θ H A = α.e A.E A = α E A (t ) cos θ + cst 4 ( ) Time dependent Schrödinger equation: i ψ (t ) = H ψ (t ) t ψ (t ) = J,ε C JJ,ε,ε e ie rot ( t t f ) / θ ϕ Raman Transitions e J, Ω,ε, M Rotational wave packet induced during the field that evolves freely in time after the laser has ended Without DC electric field: rephasing of the wave packet leads to a periodic post pulse molecular alignment With the DC electric field: Both revivals of orientation and alignment are expected. J+ J J+ J- J- Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Characterization cos θ = CO + t J,ε Mean orientation bj',ε cos(ω <cosθ > For orientation: t + φ J,J + ) J,J + Periodic oscillations at Raman frequencies t Permanent alignment = CA + bj,ε cos(ω J,J + a J,ε cos(ω J,J + J,ε + J,ε cos θ <cos θ > For alignment: t + φ J,J + ) t + φ J,J + ),6,4,, -, -,4 -,6,8,7,6,,4,3,,, Periodic oscillations at Raman frequencies In case of NO, Raman frequencies are given by ω J,J + = hbc( J + ) + κ ( J, ε ) ω J,J + = hbc( J + 3) + κ ' ( J, ε ) T = / 3Bc, / Bc, / 7 Bc... 3 3 t [ps] t [ps] J = / 8 Bc, / Bc, / 6 Bc... J = /Bc= ps /4Bc= ps J = J = Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Evolution with Intensity: Saturation.7 Saturation of both alignment and orientation after an intensity of. W /cm.6 Best value of.63 for the orientation and.78 for the alignment with a FTL pulse..4 3 4 I (. W.cm ) 6 7 aj and bj.... ω (. Hz). '... ω (. Hz) Better alignment if cos(ωj,j+ +φj,j+)= 8 for all J populated during the field cos ( ω J+,J T+φ J+,J). cos ( ω J+,J T+φ J+,J) bj <cos θ >max and <cos θ >max.8 Possible to obtain a better rephasing of the rotational wave packet using a specific pulse shape? -.. ω (. Hz). -. ω (. Hz). Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
How to manipulate the electric field Idea: The pulse profile is modified by controlling the spectral phase, with a spatial light modulator, through 8 pixels equally distributed across the spectrum, i.e. E (t ) = π + ε (ω ) e iφ (ω ) iω t Spectral amplitude e with φ(ω ) = Π n = 64 ω ωn Π φn ω n = 63 -.. Spectral phase Spectral phase controlled by modifying the phase φn of the 8 pixels Fourier Transform Limited (FTL) pulse (φn = for all n) of fs duration Modification of the pulse shape under constraint of constant energy and constant time delay Use an evolutionary algorithm to optimize the 8 phases φ(n) (genes) that maximize the impulsive orientation (<cosθ>) or the impulsive alignment (<cosθ>) Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Procedure Derandomized algorithm in a closed loop control Initial population: population 6 Initial randomly Calculation of Calculation <cosθ <cos² >(t). θ>(t). Parameters: 8 genes φn= individual φ n E (ω ) E (t ) ψ (t ) cos θ or cos θ FT TDSE New New population population: best individuals individuals cloned cloned best + new new individuals individuals (reproduction/mutation) (reproduction+mutation) Selection Selection of the best individuals individuals Evolutionary Evolutionary algorithm algorithm Convergence Convergence No yes 8 6 8 6 4 4 -. -. -. -. -. -....... The algorithm is initialized with a population of individual randomly selected (random phase values). mutations are applied to this individual and, among the new created individuals, the individual with the best fitness for the issue of alignment/orientation is chosen for the next generation. The procedure is also repeated until the algorithm converges towards some optimum for the control objective. Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Results for a state-selected NO molecules: alignment,6,4 -,8,7,6,,4,3,, - t (ps) -. ω (. Hz)... t (ps) FTL pulse : <cosθ>=.76 Shaped femtosecond pulse : <cosθ>=.94 aj and bj 6 4 3-4, I (TW/cm²) -4 <cos θ > I (TW/cm²),8 <cos θ >, cos ( ω J+,J T+φ J+,J) Energy: fs pulse of peak intensity 6. W/cm Fitness function: <cosθ> at time. ps.. ω (. Hz) All components of the wave packet are in phase at the full rotational period Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Results for a state-selected NO molecules: orientation Energy: fs pulse of peak intensity 6. W/cm Fitness function: abs(<cosθ>) at time. ps. ' 4. -3 - - ) J,J+ T+φ, J,J+, -, -, 3..4.6 ω (. Hz).8..4.6 ω (. Hz).8 t [ps], <cosθ >. bj 6 cos ( ω Intensity 8 t [ps] FTL pulse : <cosθ>=-.63 Shaped femtosecond pulse : <cosθ>=-.84 - All components of the wave packet are in phase at the full rotational period Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Experimental considerations,9,8 Need more than evaluations to reach the solution!! Experimentally, evaluation takes between 3 seconds to minute abs(<cosθ >),8,7,7 8-6 hours of experiment!!!,6,6, 3 4 Nb evaluation Idea: Use the result find by Hertz et al. : Alignment is enhanced using a sigmoid function parameterized on only threeω parameters.8 3..7 Sigmoid spectral phase. s I. φ (rad) φ (ω ) = s + exp( a(ω ω ))).6. a.4.3 -... -3. 78 79 8 λ (nm) 8 8. -. -.... t (p s) Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Simplification using a sigmoidal phase Energy: fs pulse of peak intensity 6. W/cm Fitness function: abs(<cosθ>) at time. ps For orientation 3 3. <cos θ > -. t (ps) - - FTL pulse : <cosθ>=-.63 Sigmoidal phase shaped femtosecond pulse: <cosθ>=-.77 Better rephasing that the FTL pulse but orientation smaller than the one obtained with the pulse shaped find by the evolutionary algorithm parameterized on 8 parameters Need less than generations to converge t [ps]. b'j -.. cos ( ω J,J+ T+φ J,J+) I (TW/cm )..4.6.8.4.6 ω (. Hz).8 ω (. Hz) -. Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Experimental setup First step: select molecules in one state at the center of a VMI spectrometer where molecules experience a DC electric field of 3 KV/cm using the hexapole state selector at a front of a NO molecular beam. P KD VMI Spectrometer Delay stage r se a L He e at st le r po cto xa ele s th wi or e t lin ula n d sio mo r e t isp l ligh d f 4- atia sp Second step: Pump-probe experiment: A 8 nm pump pulse is used to align and orient molecules that is probed by a 4 nm pulse via the coulomb explosion of the molecules. The D angular distribution of the fragments produced by the 4 nm pulse are recorded in a VMI spectrometer and send to a computer. Images are recorded as a function of the pumpprobe time delay. Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
(b) t=4.8ps (c) t=.3 ps (d) t= 9.ps (e) t=9.6 ps (f) t=. ps D distribution consists of spots with a relative intensity that serves as a measure of the molecular orientation.,6,4,, -, -,4,9,9,8,8,7 (b) (a) (c) (d) (e) (f),6,4,, -, -,4 -,6,8,7,6,,4,3,,, Allows to distinguish fragments with a recoil velocity away from or towards the D detector <cos θ D > <cos θ D > θd <cos θ > (a) t=4. ps <cos θ > First results: the impulsive orientation t [ps] Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Optimization of impulsive orientation (a) FTL pulse (b) Sig. phase Initial population: Initial population randomly Evaluation 7 Simplex method 4 6 3 3 4 3 3 Pixel 4,7 4 Without pump -abs(<cosθd>) Convergence Convergence Pixel 4,7 6 8 FTL pulse 4 6 Sigmoid phase pulse,6,6,,,4 The degree of orientation with the shaped laser pulse clearly exceeds the one obtained with the FTL pulse Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Optimization of impulsive orientation Sig. phase mask,4-4,6,4,, -, -,4 -,6 FTL pulse,, -, -,4,, <cosθ > <cosθ D>,6-4,8,6,4,, -, -,4 -,6 -,8 I [arb. unit],6,4,, -, -,4 -,6 t [ps] <cosθ > <cosθ D> For a long delay scan: Cross correlation Clear evidence of the optimization of the impulsive orientation by a sigmoidal phase shaped laser pulse: cos θ sig max.74 cos θ D sig max. r = = =.8 th rexp = = =.3 cos θ FTL max.6 cos θ D FTL max.38 Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden
Conclusion and outlook First observation of a strong degree of impulsive orientation using a state selected NO molecules Good agreement between theory and experiment Optimization of orientation both theoretically and experimentally Try the full optimization using experimentally the evolutionary algorithm parameterized on 8 parameters (possible to obtain experimentally <cosθ>=.84). Start with the optimization of the population transfer The unprecedented degree of molecular orientation reached in the presented study paves the way towards new applications, for instance studies of reaction dynamics in the molecular frame and orbital tomography of heteronuclear molecules Evolutionary Algorithms for Many-parameter Physics, 7- March, Leiden