Moments of Angular Distribution

There are several ways to expand the angular distribution of the wavepacket in a power series², but a natural basis consists of the Legendre polynomials, as for ∆M = 0 the eigenstates are independent of φ and the spherical harmonics simplify to Legendre polynomials. Only even order polynomials appear in the expansion since for a ground-state-selected ensemble the odd order moments describe orientation of the molecular axes,

2We had indeed originally performed the analysis in terms of squared Chebyshev polynomialscos²nθ2D for numerical convenience and the results of both approaches are identical.

4. Molecular movie of ultrafast coherent rotational dynamics of OCS

which was not present. The expansion takes on the form P(θ_2D, t) =

JXmax

k=0,keven

a_k(t)P_k(cosθ_2D) (4.2) where the full time-dependent angular distribution is denoted as P(θ_2D, t) and ak

(k = 0,2. . . J_max) are the expansion coefficients corresponding to the k-th Legendre polynomialP_k;J_max is the angular momentum quantum number of the highest populated rotational state in the wavepacket.

In order to characterize the initial state distribution of rotational states in the molecular beam, the eight lowest even-order moments of the experimental angular distributions were fitted simultaneously using least squares minimization. For each moment, squared differences were summed according to

χ²_n =X

t

(hP_2n(cosθ_2D)_expi(t)− hP_2n(cosθ_2D)_sim,voli(t))² , (4.3) where the sum runs over all measured delay times t and n= 1. . .8. In order to compute hP_n(cosθ_2D)_sim,voli(t), several steps were followed. First, the coherent wavepackets, created through the interaction with the alignment laser pulses, were for every initial state described in the basis of field-free eigenstates as

Ψ_Ji,Mi(θ, φ, t) =X

J

aJ(t)Y_J^Mi(θ, φ), aJ(t = 0) =δJ Ji, (4.4) where a_J(t) = |a_J(t)|e^iφJ(t) are time-dependent complex coefficients with amplitude

|a_J(t)|, phase φ_J(t), and initial condition a_J(t = 0) =δ_{J Ji}, δ_{J Ji} is the Kronecker delta, obtained from the solution of the time-dependent Schrödinger equation; Y_J^M(θ, φ) are the spherical-harmonic functions andJ_i, M_i are the quantum numbers of the initial state from which the wavepacket is formed. The sum runs only over J, since M was a good quantum number due to cylindrical symmetry, as imposed by the linear polarization of the alignment laser, and, hence, ∆M = 0 and M = M_i was conserved. Furthermore, the selection rules for transitions between different rotational states were ∆J = ±2, since the population transfer is achieved via non-resonant two-photon Raman transitions.

Moreover, the static VMI field was perpendicular to the alignment laser polarization and does not mix differentM states. Since more than one rotational state were initially populated, the 3D rotational density was obtained through the incoherent average with statistical weightsw_Ji,Mi

P_sim,3D(θ, φ, t) = X

Ji,Mi

w_Ji,Mip(θ)|Ψ_Ji,Mi(θ, φ, t)|² , (4.5) which were not knowna priori and used as fitting parameters. The functionp(θ) describes the angle-dependent ionization probability, which was approximated through the square of the measured angular-dependent single-electron ionization rate. Finally, a focal average over the interaction region with the alignment and probe laser beam profiles, assumed to

50

be Gaussian, was performed. The average over intensities in the laser focus was calculated through integration

hP_sim,3D(θ, φ, t)i_vol(t) = 1 N

Z rmax

0

hP_sim,3D(θ, φ, I_align(r), t)ie^{−2r2/wprobe2} rdr (4.6) with radiusr_max atI_align=1×10⁹W/cm² andN a normalisation factor. The dependence of the rotational wavepackets on the alignment laser intensities is explicitly stated in (4.6). The widths of the laser beams were also not known a priori and were included as

θ

x

y z

x z

θ_2D

Figure 4.5.: Relation between the Euler angleθ, defining the alignment of the molecular axis with respect to the pump laser polarization axis, andθ_2D, corresponding to the angle between the pump laser polarization and the detected ion-momentum distribution on the 2D detector.

further fitting parameters. The resulting focal- and initial-state-averaged 3D rotational densities were projected onto a 2D plane using a Monte-Carlo sampling routine, which included the experimental radial distribution extracted at the full revival at a delay time of 120.78 ps, yielding the simulated VMI images inFigure 4.2in the main part. The relation between the 3D rotational density and the 2D projected density is graphically illustrated inFigure 4.5. The Legendre moments of the angular distribution were then extracted from the 2D projected images and compared to experiment throughχ²_n, as described in (4.3).

The statistical weights w_JiMi of the initial state distribution and the laser focal sizes were varied until P

nχ²_n converged to its minimum. The individual populations determined through the fitting procedure are w₀₀ = 8.2·10⁻¹, w₁₀ = 3.7·10⁻², w₁₁ = 7.5·10⁻², w₂₀ = 1.5·10⁻², w₂₁ = 2.1·10⁻² and w₂₂= 3.2·10⁻² and the optimal focal parameter were determined to be walign =130 µm for the alignment laser andwprobe =60 µm for the probe laser. The results are consistent with the fact that the probe laser was tighter focused than the alignment laser such that only molecules exhibiting strong alignment, close to the beam center, are probed.

The final results of the fitting procedure are shown in Figure 4.2 in the main part and inFigure 4.6. The simulated angular distributions and the moments of the angular distribution are in excellent agreement with the experiment, in particular all oscillations are correctly captured, even for the highest-order moments. The experimental

parame-4. Molecular movie of ultrafast coherent rotational dynamics of OCS

0 25 50 75 100 125

Time (ps) 0.0

0.2 0.4 0.6 0.8

P10(cosθ2D) ^{Experiment Simulation}

0 25 50 75 100 125

Time (ps) 0.0

0.2 0.4 0.6

P12(cosθ2D) ^{Experiment Simulation}

0 25 50 75 100 125

Time (ps) 0.0

0.2 0.4 0.6

P14(cosθ2D) ^{Experiment Simulation}

0 25 50 75 100 125

Time (ps) 0.0

0.1 0.2 0.3 0.4 0.5

P16(cosθ2D) ^{Experiment Simulation} h

0 25 50 75 100 125

Time (ps)

−0.25 0.00 0.25 0.50 0.75 1.00

P2(cosθ2D) ^{Experiment Simulation}

0 25 50 75 100 125

Time (ps) 0.0

0.2 0.4 0.6 0.8 1.0

P4(cosθ2D) ^{Experiment Simulation}

0 25 50 75 100 125

Time (ps)

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

P6(cosθ2D) ^{Experiment Simulation}

0 25 50 75 100 125

Time (ps) 0.0

0.2 0.4 0.6 0.8

P8(cosθ2D) ^{Experiment Simulation}

e f

g

a b

c d

Figure 4.6.: Even order moments 1 to 8 of the angular distribution a hP₂(cosθ_2D)i, b hP₄(cosθ2D)i, c hP₆(cosθ2D)i, d hP₈(cosθ2D)i e hP₁₀(cosθ2D)i, f hP₁₂(cosθ2D)i, g hP₁₄(cosθ_2D)i,h hP₁₆(cosθ_2D)i.

ters used for the simulations were the peak intensities for the two alignment pulses of I_align,1 =1.92 TW/cm² andI_align,2=5.5 TW/cm², the pulse duration of the alignment laser pulsesτ_align =255 fs, the time delay between the two alignment laser pulses τ_delay =38.1 ps, and the pulse duration of the probe laser τ_probe =60 fs. Calculations with 21 initial states, i. e.,J = 0. . .5, M = 0. . .5, included in the initial rotational state distribution were originally performed, but convergence was already reached for the 6 lowest-energy states and the fitting procedure was restricted to using these 6 lowest rotational states, i. e., J = 0. . .2, M = 0. . .2, and the focal volume was averaged over 100 intensities in I_align = 1·10⁹. . .5.5·10¹²W/cm². In all calculations the basis for each coherent wavepacket included all rotational states up toJ = 50.

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Im Dokument Molecular-Frame Angularly-Resolved Photoelectron Spectroscopy (Seite 69-73)