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