ultrafast molecular imaging by laser induced electron diffraction [0.4
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Ultrafast molecular imagingby laser induced electron diffraction
– LUMAT – Journee Theoriciens –
Eric Charron
ISMOUniversité Paris-Sud
November 7, 2012
Eric Charron (Orsay) Molecular imaging November 7, 2012 1 / 16
Acknowledgements
Osman Arne Raiju
Tung Michel Christian
Eric Charron (Orsay) Molecular imaging November 7, 2012 2 / 16
Ultrafast molecular imaging
OUTLINE
Introduction & theoretical model
Analysis of the electron momentum distribution
Applications to CO2 : effect of orbital symmetry
Orbital tomography
Conclusion & outlook
PRA 83, 051403(R) (2011) – PRA 85, 053417 (2012)Eric Charron (Orsay) Molecular imaging November 7, 2012 3 / 16
Introduction
Recollision mechanism
ε (t) Linear polarization800 nm / 1014 – 1015 W.cm−2
t
V(r) + e r . ε (t)Coulomb Interaction
P.B. Corkum PRL 71, 1994 (1993)
Eric Charron (Orsay) Molecular imaging November 7, 2012 4 / 16
Introduction
Comparison with conventional imaging techniques
I X-ray (solid state)I Electron diffraction (gas phase)
I Spatial resolution on the order of Å or less.I Temporal resolution > time-scale of atomic motion.
Alternative approach : Laser induced electron diffraction
Emax ' 3.17Up = 3.17× e2ε20
4mω20' 100 eV @
{800 nm ; 1015 W/cm2
}λDB ' 1 Å
I Drawback : Requires pre-aligned moleculesI Advantage : Femtosecond resolution < time-scale of atomic motion
Eric Charron (Orsay) Molecular imaging November 7, 2012 5 / 16
Theoretical model : co2 – sae, 2d, fixed nuclei
TDSE : i h∂tΨ(~r, t) =[H0(~r |R) + Vint(~r, t)
]Ψ(~r, t)
Field-free Hamiltonian : H0 = − h2
2m∇2~r + Veff(~r |~R)
with :
Veff(~r |~R) =
∑j∈nuclei
−Zj(~r |~Rj)√|~r− ~Rj|2 + aj2
Zj(~r |~Rj) = Z∞j + (Z0
j − Z∞j ) exp(−|~r− ~Rj|
2/σj2)
Interaction :
Vint(~r, t) = e~r .~ε(t)
~ε(t) = ~ε0 f(t) cos(w0t+ϕ)
Ψ(~r, ti) −→ Ψ(~r, tf) : Split-operator method (~r /~k)
Eric Charron (Orsay) Molecular imaging November 7, 2012 6 / 16
Wave packet dynamics
t
V(r) + e r . ε (t)Coulomb Interaction ε (t)
P.B. Corkum PRL 71, 1994 (1993)
Linear polarization800 nm / 1014 W.cm−2
CO2 : Highest occupied molecular orbital (HOMO)
C
O
O
ε (t)
Eric Charron (Orsay) Molecular imaging November 7, 2012 7 / 16
Asymptotic analysis
Electron momentum distributions
P(~k) ∝
∫ei
~k .~r Ψas(~r, t→∞) d~r
O
C
O
kx
(a.u
.)ky (a.u.)
P(~k)
−→
kx (a.u.)
S(kx) =
∫P(~k)dky
S(kx)
Measure of RCO = π/∆k
Accuracy ' ±0.05 Å
Eric Charron (Orsay) Molecular imaging November 7, 2012 8 / 16
Effect of orbital symmetry : the facts
-13.8 eV
-17.3 eV
Energy HOMO P(~k)
HOMO-1 P(~k)
1.0 1.5 2.0 2.5 3.00.00
0.01
0.02
0.03
0.04
0.05
0.06
S(kx)
kx (a.u.)
1.0 1.5 2.0 2.5 3.00.00
0.02
0.04
0.06
0.08
0.10
S(kx)
kx (a.u.)
Eric Charron (Orsay) Molecular imaging November 7, 2012 9 / 16
Effect of orbital symmetry : the analysis (1)
TDSE : i h∂tΨ(~r, t) =[H0(~r |R) + Vint(~r, t)
]Ψ(~r, t)
Formal solution : Ψ(~r, t) = U(t← ti)Ψi(~r)
where : i h∂tU(t← ti) =[H0(~r |R) + Vint(~r, t)
]U(t← ti)
Dyson : U(t← ti) = U0(t← ti) −i h
∫tti
U(t← t ′)Vint(~r, t ′) U0(t′ ← ti)dt
′
Solution : Ψ(~r, t) = Ψi(~r, t) − i h
∫tti
U(t← t ′)Vint(~r, t ′)Ψi(~r, t ′)dt ′
Transition amplitude :
a(k, tf) = − i h
∫tti
⟨Ψk(~r)
∣∣U(t← t ′)Vint(~r, t ′)∣∣Ψi(~r, t ′)
⟩dt ′
Eric Charron (Orsay) Molecular imaging November 7, 2012 10 / 16
Effect of orbital symmetry : the analysis (2)
Approximations :
{I SFA : U(t← t ′) ' UV(t← t ′)
I PWA : Ψk(~r) ∝ Φk(~r) = e−i~k·~r
where UV(t← t ′) is the time evolution propagator associated with
HV(~r, t) = − h2
2m∇2~r + Vint(~r, t)
This yields an approximate transition amplitude :
aSFA(k, tf) ' ie h
∫tti
e−iS(k,tf,t′,ti) E(t ′) 〈Φk′(~r) |y |Ψi(~r)〉 dt ′
where k ′ = k − e h[A(tf) −A(t ′)]
With E(t) ∼ E0 cos(ωt), we get A(t) ∼ E0ω
sin(ωt)
And therefore k ′ ' k for |k| > eE0 hω
Eric Charron (Orsay) Molecular imaging November 7, 2012 11 / 16
Effect of orbital symmetry : the analysis (3)
And finally, for |k| > eE0 hω
:
aSFA(k, tf) ' ie h〈Φk(~r) |y |Ψi(~r)〉
∫tti
e−iS(k,tf,t′,ti) E(t ′)dt ′
And therefore : PSFA(~k) ∝∣∣∣F[yΨi(~r)
]∣∣∣2
And : SSFA(kx) =
∫ ∣∣∣F[yΨi(~r)]∣∣∣2dky
Physical interpretation :
{I Direct ionization.
I Large electron momentum.
Eric Charron (Orsay) Molecular imaging November 7, 2012 12 / 16
Effect of orbital symmetry : the analysis (4)
HOMO :
Ψi(~r) ∝ 2pO(x,y+ R) − 2pO(x,y− R)
where 2pO(x,y) ∝ x exp(−αr)
SSFA(kx) ∝sin2(kx R)
(k2x + α2)9/2
1.5 2.0 2.5 3.0
kx (a.u.)
0.00
0.02
0.04
0.06ExactSFA
Reconstructed HOMO
α ' 2.1 a.u.Eric Charron (Orsay) Molecular imaging November 7, 2012 13 / 16
Effect of orbital symmetry : the analysis (5)
HOMO-1 :
Ψi(~r) ∝ 2p(x,y+ R) + γ 2p(x,y) + 2p(x,y− R)
SSFA(kx) ∝[γ+ 2 cos(kx R)]
2
(k2x + α2)9/2
1.5 2.0 2.5 3.0
kx (a.u.)
0.00
0.02
0.04
0.06 ExactSFA
Reconstructed HOMO-1
Eric Charron (Orsay) Molecular imaging November 7, 2012 14 / 16
Effect of pulse duration
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
S(kx)SSFA(kx)
kx (a.u.)
HOMO – Pulse duration : 10 fs
Eric Charron (Orsay) Molecular imaging November 7, 2012 15 / 16
Conclusion & outlook
Molecular imaging with LIED : Advantages / Constraints
I Yields an « instantaneous » (fs) snapshot of the moleculeI Robustness : vibr. distribution ; pulse duration (1-10 opt.cycles)I Possibility to follow nuclear dynamics : « RCO(t) »I Destructive measurementI Molecular information carried by high-energy electronsI Requires to pre-align the molecule : ∆θ ∼ 20˚
Perspectives
I Field polarization / Measurement of bond anglesI Non-linear polyatomic moleculesI Test the measurement of electronic dynamicsI Electron correlation effects
Eric Charron (Orsay) Molecular imaging November 7, 2012 16 / 16