1. typical kHz experiment
amp
tdc
TOF/MS
TMP
UHV
time
-metal
faraday
photodiode
disc

2. [o  int(t)](t)  iħ(t) 
high sensitivity results
HHG
electrons
photoelectron
total rate
[o  int(t)](t)  iħ(t) 
TDSE-SAE 10 20 30
TW/cm2 20 15 10
TW/cm2
xenon, 1m, 30ps

3. plat du jour: helium & the rebirth of the classical picture
helium: kHz experiment
0.8 m
1 PW/cm2
simpleman
plat du jour: helium & the rebirth of the classical picture

4. strong-field atomic physics II
Louis DiMauro
log(electrons)
electrons
1 PW/cm2
time-of-flight (s)
ions +7 +5 +3 +1
argon
charge
states
Lompre et al. (Saclay) He
800 nm
1.6 eV
wavelength (nm) 16 14 10 12 8
log(photons)
H51
H79
photons
High harmonic generation
L’Huillier et al. (Lund)
Non-linear
Non-resonant
Non-perturbative

5. why helium? largest binding energy (Ip = 24.5 eV) of all neutral atoms
total rate
B. Walker et al., PRL 73, 1227 (1994)
theory: He  He+ + e
ADK
TDSE-SAE
OTB
over-the-barrier (OTB) ionization: Eo = Ip2/4q3Z helium: Eo = 0.2 au (1.4 PW/cm2)
R  I3/2
measurements: Is = 0.8 PW/cm2 (Eo = 0.15 au)
Keldysh:  = (Ip/2Up)1/2 = 0.49 Up = m  = 50 au (25Å)  ao ~ 1Å

6. the simpleman’s picture of ionization
quasi-classical description o
electric field E = Eo sint
Field amplitude  2
Time 
velocity v(t) = Eo/[cost - coso] + vo
quiver drift
for tunneling, vo=0
Two Step Approach  Excitation  Field Motion (Continuum)

7. classical model: rescattering
x(t) = Eo/2 (sint - sino + (o - t) cos o)
Optical Cycles
electron-core interaction ~ ½ cycle
electron gains field energy
Position
collision 2 1 3
phases  collision trajectories
Optical Cycle
simple symmetry argument~ ½  contribute
Position
Time 3 5 
Position
Time 3 5 
Position
Time
Re-examine the trajectories for collisions
½ return to the core!!

8. classical model: rescattering
Schafer, Yang, DiMauro & Kulander PRL 70, 1599 (1993)
P. Corkum PRL 71, 1994 (1993)
excitation
t=0
propagation
¼-cycle
rescattering
½-cycle
time 
Invoke a Wave Packet View at the End.
3-step view of quasi-classical rescattering.

9. classical model: high harmonic connection
HIGH HARMONIC RADIATION
Helium, 800 nm
Cutoff ~ 3Up + IP  3Up
gas jet
gas filled capillary
Murnane & Kapteyn
800 nm
25 fs
1015 W/cm2
HHG
log (number of photons)
helium 16 14 12 10 8
wavelength (nm)
H79
H51
table-top source of coherent short wavelength light.
potential for generating attosecond (10-18 s) light pulses.

10. classical model: rescattering
PHYSICAL CONSEQUENCE:
electron capture results in odd harmonic photons.
harmonic cutoff: (3Up + IP) rule !!
elastic scattering yields energetic (10Up) electrons.
inelastic e-2e scattering  multiple electron ejection.
excitation
t=0
propagation
¼-cycle
rescattering
½-cycle
time 
3.17 Up cutoff
return energy: T(t = tr) = 2Up (cos2 r + cos2 o - 2cos o cos r )
Cutoff rule:
3.17Up + Ip

11. quantum model: TDSE-SAE
elastic rescattering
tunnel (vo=0) v(t) = Eo/[cost - coso]
backscatter ( = ) set: v(r) = -v(r)
v(t > tr) = Eo/[(cost - cosr) – (cosr - coso)]
initial bs velocity
normal drift
new elastic cutoff:
T = 10Up
quantum model: TDSE-SAE
K. Schafer et al. PRL 70, 1599 (1993)
Describe curves and highlight double rate
Prompted work by many theory groups
Emphasizes the power of kilohertz

12. elastic rescattering: SFA approximation
Semi-classical solution of generalized SFA
Lewenstein et al., PRA 51, 1495 (1995)
backscattering results in production of high energy electrons

13. rescattering: numerical method (Kulander)
divide optical cycle into a large number of equally spaced time intervals and calculate a tunneling rate.
at each phase of the field, launch a gaussian wave packet at the outer turning point of the suppressed effective potential with zero velocity.
initial conditions are determined from SAE results.
propagate in the combined field until it escapes or returns to the plane of the nucleus.
returning trajectories are assumed to spread freely.
allow for only one return of the wave packet.
calculate the differential elastic cross-section using partial wave analysis.
electron spectrum is determined by summing all time intervals.
double ionization is calculated using field-free e-2e inelastic cross-section.
spatial and temporal averaging is included for comparison to measurement.

14. elastic rescattering: experiment & theory
helium, 0.8 m, 0.8 PW/cm2
the short-range physics is important.
quantum diffusion reduces the effective rescattering.
the recollision occurs in less than an optical cycle.
Make comparison with Helium spectrum
Describe the Quantum Calculation
Describe all the PES studies performed

15. elastic rescattering: intensity dependence
helium, 0.8 m
Remember, Up  Intensity !!
PW/cm2
in scaled energy, distributions look similar!

16. elastic rescattering: intensity dependence
Rutherford (coulomb) scattering  b vo
 3Up  

17. elastic rescattering: intensity dependence
Rutherford predicts a 100-fold decrease in high energy electrons over intensity range of experiment.
not bad for an experimentalist.

18. elastic rescattering: intensity dependence
helium: experiment & theory
energy (eV) Up
coulomb
e – He+

19. elastic rescattering: atomic dependence
rescattering is sensitive to the atomic core (cross-section).
1 PW/cm2
coulomb
e – He+
e – Ne+
intensity (W/cm2)

20. elastic rescattering: differential cross-section b vo
initial bs velocity
normal drift
vx(t) = Eo/[(cost - cosr) – cos (cosr - coso)]
vy(t) = -Eo/[sin (cosr - coso)]
Theta=0, vy=0
Theta=pi, vx=max
Correlation between scattering angle and final energy
relationship between scattered, , and detected, d, angles.

21. elastic rescattering: differential cross-section
electron cutoff energy versus detector angle
8Up
6Up
2Up

22. elastic rescattering: differential cross-section
helium, 0.8 m, 1 PW/cm2

23. elastic rescattering: differential cross-section
helium: experiment & theory
electron counts

24. quantum view
Ken Kulander

25. strong-field double ionization
time 
 m/q
“direct”
ionization
multiple charge states readily observed in an intense laser field.
some charge states cannot be described by a “single” rate.

26. helium double ionization: total rate
two mechanisms result in the formation of He2+ !!
He  He+ + e
He+  He2+ + e
some insights into double ionization:
NS linked to depletion of the neutral ground state.
first electron tunnels into the continuum.
the NS yield is strongly polarization dependent as compared to the sequential processes.
Describe curves and highlight double rate
Prompted work by many theory groups
Emphasizes the power of kilohertz

27. we ran out of steam: computationally
tomorrow’s plat du jour
two-electron soup á la carte
experiments pioneer the future

28. helium double ionization: polarization dependence
0.2 (NS) and 4 (sequential) PW/cm2
In neon, polarization dependence of NS and HHG agreed with classical analysis
Dietrich et al. PRA 50, R3585 (1994)

29. helium double ionization: high sensitivity
Experiment performed at two intensities.
0.8 PW/cm2 1/500
0.4 PW/cm2 1/1000
3He is used for coincidence measurement.

30. helium double ionization: high sensitivity 1800 “background” electrons
2 “signal” electrons
The Needle in the Haystack

31. helium double ionization: e-ion coincidence
interaction
region
e spec
mass spec
mechanical referencing design
common interaction volume
pulsed mode operation
dual MCP detection
UHV environment (10-10 t)

32. helium double ionization: e-ion coincidence
mass spectrometer
electron spectrometer

33. e-ion coincidence apparatus: test
an 8:1 Xe:Kr gas mix test was used to test the coincidence apparatus.
T:F ~ 3:1
it really, really works!

34. helium double ionization: electron distributions
4×1014 W/cm2
205M shots
45M He hits
1058 He2+ coin
He+
He2+
8×1014 W/cm2
double ionization results in “hotter” distribution than single ionization.
distribution consistent with e-2e rescattering.

35. helium double ionization: classical interpretation
electron energy (Up)
counts (arb units)
100 1
.001 2 3 5 4
backscattered
forward
release first electron at phase i
if return energy is sufficient to excite second electron to first excited state (40 eV), then proceed.
all excess energy goes to first electron (forward or backward).
second electron is then field ionized with zero initial kinetic energy.
8×1014 W/cm2
e-2e
Corkum (1993)

36. helium double ionization: S-matrix calculation
shake-off
correlated energy sharing

37. double ionization experiments: other atoms
Frankfurt group Ar2+ & He2+ ion recoil (COLTRIMS) Ar2+ e-ion coincidence
Freiburg group Ne2+ ion recoil (COLTRIMS) Ar2+ e-COLTRIMS
Crete group Xe2+ e-ion coincidence
Michigan group Ar2+ e-ion coincidence
BNL group Ar2+ & Xe2+ e-ion coincidence

38. is everything perfect in the world?
helium, 0.4 m
reduce ponderomotive energy by 4 since Up  2
e-2e
classically
forbidden NS
The double-to-single ionization ratio is equal for 800 nm & 400 nm excitation.