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VOLUME
71,
NUMBER
13
PHYSICAL REVIEW LETTERS
27
SEPTEMBER
1993
Plasma
Perspective
on Strong-Field
Multiphoton
Ionization
P.
B. Corkum
National
Research Council
of
Canada,
Ottawa, Ontario, Canada
K1A
OR6
(Received
9
February
1993)
During
strong-field
multiphoton ionization, a wave
packet
is formed
each time the laser field
passes
its
maximum
value. Within the
first laser
period
after ionization there is
a
significant
probability
that
the
electron
will
return
to the
vicinity
of the ion
with
very
high
kinetic
energy.
High-harmonic
generation,
multiphoton
two-electron
ejection,
and
very
high
energy
above-threshold-ionization
electrons
are all
consequences
of this
electron-ion
interaction.
One
important parameter
which
determines the
strength
of these effects is
the rate at which
the wave
packet spreads
in
the direction
perpendicular
to the
laser
electric
field;
another is the laser
polarization.
These will
be crucial
parameters
in future
experiments.
PACS numbers: 32.80.
Rm
This
paper
introduces
a
model
for
nonperturbative
non-
linear
optics
involving
continuum states.
The model
sug-
gests ways
of
optimizing
and
controlling
the high-order
nonlinear
susceptibility.
In addition,
it clarifies the
con-
nection between
above-threshold
ionization and harmonic
generation.
For
example,
the
model shows
that
there
is not a
one-to-one
correspondence
between
above-
threshold-ionization
peaks
and
harmonic emission. It
also sho~s
why
the maximum
energy
of
the harmonic
emission
is
very
diA'erent
from
the
maximum
above-
threshold-ionization
energy.
However,
the
strength
of
the
harmonic
emission increases
approximately
linearly
with
the ionization rate.
The
paper
applies
concepts
long
familiar
to
plasma
physicists
to
strong-field atomic
physics.
Since
the
prod-
ucts of atomic ionization
are the basic constituents of
plasmas,
we should
not
be
surprised
if
plasma
methods
are
applicable.
High-field
atomic
physics
and
plasma
physics
are
becoming
increasingly
entwined
[1].
The essential
point
in
the
paper
is that an atom that
undergoes
multiphoton
ionization does not
immediately
become
a
well-separated electron and ion.
Rather,
there
is a
significant
probability
of
finding
the
electron in
the
vicinity
of the
ion
for one or
more laser
periods.
This pa-
per
extends
the
quasistatic
model
of multiphoton
ioniza-
tion
[2,
3]
to include
the
electron interaction
with the ion.
By
doing
so,
the
model
is able
to
quantitatively
predict
double ionization
[4],
hot
above-threshold ionization
[5],
and,
most
important,
high-harmonic
generation
[6,
7].
The
paper
presents
a
unified
approach
to
all three
phe-
nomena. This
is
done
using
only
one
free
parameter
that
is
severely
constrained. Furthermore,
the
parameter
is
subject
to
independent
experimental
and
theoretical
study.
The
quasistatic
model,
as it
has
been
applied,
consists
of
a
dual
procedure
which will
now be
outlined.
First,
one determines the
probability
of
ionization
as a
function
of the
laser electric field
using
tunnel ionization models.
For all
calculations in this
paper,
the
ionization rate is
given
by
[8]
Mac
-
tp.
l
C,
~
t.
I
'Gtm
(4a),
/to,
)
'" '
exp(
—
4to,
/3to,
),
(1)
~here
ta,
=E,
/h
co,
=e@(2m
E
)
''
n*=(E"/E')'1'
G
=(2!+1)(l+
(m
()!(2
!
~)/)m
~!(1
—
)m))1,
and
~C„
t
~
=2
"
[n*I
(n*+1
+1)I
(n
—
1*)]
'.
In
Eq.
(1),
E,
is the
ionization
potential of
the
atom
of
interest,
E,
is the
ionization
potential of
hydrogen,
l and m are
the
azimuthal
and
magnetic
quantum numbers, and
6
is
the
electric field
amplitude.
The
effective
quantum
number
l is
given
by
I
=0
for 1«n
or
l
=n
—
1 otherwise.
The
probability
of
ionization,
P(t), during
the time
inter-
val'
dt
is
P(t)
=Wa,
(8(t))dt,
where
C(t)
is
the magni-
tude
of
the electric field
8
(t)
=
8
pcos(tot
)e„+
a6'p
xsin(tot)e~.
The
tunneling
model
describes the
forma-
tion
of a
sequence
of wave
packets,
one near each
peak
of
the
laser electric
field.
The second
part
of the
quasistatic
procedure
uses
clas-
sical mechanics to
describe the
evolution of
an electron
wave
packet.
For
siinplicity,
we
shall consider
only
the
electric
field of the
laser. Both the
magnetic
field
of
the
laser
and the
electric field
of the
ion,
for
example,
are ig-
nored.
The
initial
conditions of
velocity
and
position
equal
0
(the
position
of
the ion) at the time
of ionization
have been
justified
previously
in the
long
wavelength
limit
[2]
by
the
comparisons
with above-threshold-ionization
experiments.
After
tunneling,
the
electron motion
in the
field
is
given
by
x
=xp[
—
cos(cot)]+vp„t+xp
y
=axp[
—
sin(tot)]+avpst+yp~,
(2)
v„=
vp
sin(toi
)
+
vp„,
v~
=
—
av
pcos(tot )
+
vp~,
(3)
where
a=O
for
linearly
polarized
light
and
a= 4-1
for
circular
polarization;
vp=q8p/m, cp,
xp
qt p/m,
to,
and
Uoz
Uoy
xone
and
y0~
can be
evaluated
from the initial
conditions
(position
and
velocity
equal
to
0
at
the time
of
tunneling).
The
energy
associated
with
the
velocities
vp„
and
Uo~
constitute
the
above-threshold-ionization
energy
in ultrashort
pulse
experiments
[2].
For
circularly
polar-
ized
light,
Eq.
(3)
indicates
that
the electron
trajectory
never
returns
to
the
vicinity
of the ion.
Consequently,
electron-ion
interactions will
not be
important.
This
gives
an
opportunity
to
test
this
part
of
the
model before
proceeding.
0031-9007/93/71
(1
3)/1994(4) $06.
00
1993 The American
Physical
Society
VOLUME
71,
NUMBER
13
PHYSICAL
REVIEW
LETTERS
27 SEPTEMBER
1993
Recent
experiment
[9]
and
analysis
[10]
indicate
that
the
quasistatic
predictions
are at
least
approximately
val-
id for
intensities
as
low as those
having
a ponderomotive
energy
equal
to
the
ionization
potential.
Since
the
ul-
trashort
pulse
above-threshold-ionization
spectrum
for
circularly polarized
light
is
completely
determined
by
Eqs.
(1)
and
(3),
precise
verification of
the
quasistatic
approach
is
possible.
Because
of
space
limitations,
we
can
only
summarize
these
findings
here.
There is
excel-
lent agreement
between
the
quasistatic
predictions
and
experimental
[5]
results
published
for
above-threshold
ionization of
helium
using
0.
8
pm
circularly
polarized
light.
It
is
clear
that
Eq.
(1)
must accurately
predict
the
sequential
ionization rate.
Such
accurate
predictions
are
essential
for
what follows,
especially
for
calculations
of
the
correlated
two-electron
multiphoton
ionization
rates.
Having
established
the
accuracy
of
Eqs.
(1)
and
(3),
we
now
discuss the
implications
for
linearly
polarized light.
Equations (1)
and
(2)
show that
half of
the electrons
that
are
field
ionized
by
linearly
polarized light
pass
the
position
of the
ion
(x-0)
once
during
the first
laser
period
following
ionization.
The other
electrons
will
nev-
er
pass
the
position
of the
ion.
Equations
(1)-(3)
deter-
mine
the
probability,
P(E),
per
unit
energy
per
laser
period
of
finding
an
electron
passing
the ion
with
energy
E.
Figure
1
shows P(F.
)
obtained
assuming
uniform
il-
lurnination,
5X10'
W/cm,
800
nm
light
interacting
with helium.
The
most
likely
and
the
maximum velocity
of
an
electron
passing
the
nucleus corresponds
to
an
in-
stantaneous
kinetic
energy
of
3.
17
times the
ponderomo-
tive
potential (3.
17U~).
(As
we
shall
see
below,
this is
the
physical
origin
of
the 3.
2U&+F,
law
for the
high-
harmonic
radiation
cutoff
[6].
)
An
electron
ionized
by
tunneling
at mt
=17,
197,
etc.
,
will
arrive at
the ion
with this
velocity. Clearly
it
is
inappropriate
to
ignore
the
interaction
between
the ion
and
this
returning
elec-
tron. We now
discuss three
aspects
of
this interaction.
One consequence
of the
electron-ion interaction can be
immediately
understood. If
the
energy
of the
electron as
it
passes
the ion exceeds the
e-2e
scattering
energy,
the
ion
can
be collisionally
ionized
by
the electron
that has
left
the
atom
only
a
fraction of
a
period
earlier.
In other
words,
correlated
two-electron
ejection
should be
ob-
served.
Figure
2
shows
the
calculated ion
yields
obtained
as a
function
of the
laser
intensity
for
0.
6
pm
light
in-
teraction
with helium.
The
agreement
with
published [4]
experimental
results
is
extremely
good.
To
obtain
this
curve,
the
known collision
cross section of
He+
was used
[11].
The
only
free
parameter
in
the
model
was the
transverse
spread
of the
electron
wave function,
or,
equivalently,
the
range
of
possible
impact
parameters.
For
the
data
shown in
Fig.
2 the
wave function
was
as-
sumed to
have
a
Gaussian
probability
distribution
for
the
impact
parameter
with
a
half
intensity
radius
of
1.
5
A.
Because of
the presence
of
the
strong
laser field,
inelastic
scattering leading
to
excited states should
also
contribute
to the
experimental
results
since an
atom in an
excited
state
should
immediately
ionize.
If
inelastic scattering
is
included,
the
agreement
is
also
good,
provided
that
the
radius is
increased
to
-2
A. Even at
2
A,
the
transverse
spread
of
the wave
function is
less
than
estimated
previ-
ously
[2]
by
comparing
the
long
wavelength
limit of
Reiss
ionization
model
[12]
with the quasistatic
predic-
tions.
The
origin
of
this
discrepancy
is
unclear.
It
may
be
the result
of
neglecting spin
correlation
effects
[13].
The
electron can
also
scatter elastically.
Any
electron
that
scatters is
dephased
from
its
harmonic
motion
and
therefore
absorbs
energy
from
the
field. Following
the
same
approach
used
to
study
inverse
bremsstrahlung
in
plasma
physics
[14],
we
assume that
tan(g/2)
-P/P„
where
g
is
the
angle
of deviation
of
the
electron,
P
is
the
Q7
0)
C3
C3
)0'
CU
O
CL
50
100
&5O
&o"
Intensity (W/cm2)
)
@16
Electron
Energy
(eV)
FIG. 1.
Velocity
distribution
for
electrons
at the
time of their
first
encounter
with
the
ion. The
parameters used
for this
cal-
culation
were
those
of helium
with
light
intensity of
5X10'
W/cm
and
wavelength 0.
8
pm.
The
sharp
cutoff'
in the
elec-
tron
energy
occurs at
3.
17U~.
FIG.
2. lon
yield
of
singly
(left curve)
and
doubly
(right
curve)
charged
ions
plotted as
a
function of
the
peak
laser
in-
tensity. The
parameters were chosen to
match those
in
Ref.
[4],
that
is,
ionization of helium
with
a
—
100
fs
pulse
of
linearly
po-
larized
0.
6
pm
light
and
a Gaussian
spatial
and
temporal
profile.
1995
VOLUME
71,
NUMBER
13
PHYSICAL
REVIEW
LETTERS
27 SEPTEMBER
1993
impact
parameter,
and
P,
is the
critical
impact
parameter
given
by
P,
=q
/4zsom,
v,
with
v
the
electron
velocity
as
it
passes
the ion.
This
equation,
which
describes
field-free
scattering,
should be
a
good
approximation
for
large
elec-
tron
velocities
and
large impact
parameters.
Figure
3
was
obtained
by
assuming
that
both
the
elastic
and
in-
elastic
scattering
angle
could be
approximated
by
the
elastic
scattering angle.
For each moment of
ionization
t',
the time
t,
velocity
U,
and
probability
of
the
electron
passing
the ion
were determined
using
Eqs.
(1)-(3). In
the case of inelastic
scattering,
the scattered
electron was
assumed
to
lose sufficient
energy
to
account
for the
ion-
ization
potential
of
He+.
The
other
electron was
as-
sumed
to be
produced
with zero
kinetic
energy.
The
elec-
tron
velocity
after
scattering
was
used as the
initial
condi-
tion in
Newton's
equation
to
determine the final
above-
threshold-ionization
spectrum.
The
parameters
used
in
Fig.
3
were
those
reported
for
published [5]
above-
threshold-ionization
spectra
obtained for helium
using
0.8
pm
light.
Both the
experiment
and the model
show
that
electrons with
energy
much
greater
than the ponderomo-
tive
energy
(
—
100
eV) at the
saturation
intensity
for
helium are
produced. Thus,
there is
qualitative, although
not
quantitative,
agreement
between
calculation and
ex-
periment.
With
the use of
diA'erential
scattering
cross
sections
for
helium, the
model
results
can
be
improved.
Indeed
they
should also
allow
predictions
of
the
energy
and
angular
dependence
of correlated
electron produc-
tion.
Another
consequence
of the
electron-ion
interaction
is
the emission
of
light.
!f the
ground
state is
negligibly
depleted,
the
wave
packet
will
pass
the ion in the same
way
during
each
laser
cycle. Thus,
any
light
that is
emit-
ted will
be at a harmonic of the laser
frequency.
The emission
can be calculated
from
the
expectation
value of the
dipole
operator
(pierian).
If
we
assume that
y=yg+y„where
yz
is the
ground
state wave function
O
10
CD
LU
10
O
JD
cD
1
0-
E
1
0-5
CD
)
106
cU
a-
10
0
200
400
Electron
Energy
(eV)
and
y,
is
the
continuum
wave
function,
then
the
dipole
moment
can
be rewritten
as
(pierian)
=(ygieriy,
)
+(y,
ieriy,
)+c.
c.
To
evaluate
this
integral,
the follow-
ing
simplifications
were
made:
(1)
The
ground
state
was
negligibly
depleted
and was
approximated
by
the
ground
state wave
function
of
hydrogen.
With
this
assumption,
(ygieriy,
)+c.
c.
are
the
dominant
terms and
account
for
the
high-harmonic
radiation.
(2) The
continuum
wave
function
was
constructed
using
the
correspondence
princi-
ple.
Since
harmonic
radiation
of
a
given
harmonic
fre-
quency
(Ep,
/h)
must
come
from
electrons
in
an
energy
range
Ep
—
hro (E
(Eg+hru,
it is
convenient
to
write
the
wave
function
as
it
passes near the
origin
as
FIG.
3.
Above-threshold-ionization
electron
energy
spec-
trum.
The
relative
number
of
electrons
are
plotted
as
a
func-
tion of
their
energy.
The
calculations
used
parameters
to match
those in the
experiment
reported
in
Ref.
[5],
that
is,
ionization
of
helium
with
0.
8
pm
linearly
polarized
light
with
a
peak
in-
tensity
of
4&10'
W/cm2.
A Gaussian
spatial
and
temporal
profile
was
assumed
with
a
fu11-width-at-half-maximum
pulse
duration
of
—
100
fs. For
reference,
the
spectrum
with
the
elec-
tron
scattering cross
section
of 0
is shown
(left curve).
y,
(x
=O,
t)
=QAp(x
=0
t)exp[ip&(x
=
O,
t)x/hlexp[
—
i
[pi
(x
=O,
t)
/(2m,
+E,
)t/hl],
h
where
the
index
of
the sum
labels
the
harmonic
and
ph(x
=
O,
t)
is
the
electron
momentum
that will
lead
to
a
given
harmonic
(Az
is defined
below).
(3)
An
electron
born
in
the
phase
interval
mt
&
17'
can
have
the
same
energy
passing
the
ion
as
one
born
in the
interval
cot
&
17'.
These
contributions
to
the
harmonic
emission
were
added
incoherently.
(4) To
obtain
the normaliza-
tion
parameter
Az,
the
transverse
spread
r
of
the
electron
wave
function
was
assumed
to
be
linear
in
time
with
a
magnitude of
—
1.
5
A/fs. This
spread
is
consistent
with
the two-electron
ejection
calculations
(Fig.
2). The
wave
function
spread
in
the
direction
of
propagation
was
taken
as the
electron
velocity
[p(x
=0)/m,
]
multiplied
by
the
time
diAerence
Bt
between
the
times
when
electrons
of
energy
Eh
—
@co
and
Eh+
Aco
pass
the
nucleus.
The
nor-
malization
condition
was
r
E~+hco
~
A~d
x=
J~
~
P(E)dE
V,
where
V=zr
pBr/m,
Figure
4
shows
the
calculated
harmonic
spectrum
of
the
absolute
value
of
the
dipole
moment
squared,
aver-
aged
over
one
period
and
measured in
atomic units. The
calculation
was
performed
for 1
pm
light
interacting
with
helium.
The
parameters
were
chosen
to be the
same as
those
used
in
a recent
Schrodinger
equation
simulation of
high-harmonic
generation
[6].
Figure
4
bears
a
remark-
able
similarity to
the results
of the
simulation. The
pla-
teau
region
has
the
same
structure.
They
both
begin
at
—
10
atomic
unit
and
decay
in
an extended
plateau
un-
til
a
photon
energy
of
—
225
eV,
a
value
equal
to
3.
17
times the
ponderomotive
energy
+
the ionization poten-
1996
VOLUME
71,
NUMBER
13
PHYSICAL
REVIEW
LETTERS
27
SEPTEMBER
1993
1p
)
p-7
)0
-9
tp
)
p-11
25
'1
25
225
tial. Even the
magnitude
of
the
high
harmonics
agrees
within
less than an order
of
magnitude.
It is clear
from
this
agreement
that the
quasistatic
model catches the
essence
of
high-harmonic
generation.
Because
of
the
simplicity
of the
quasistatic
model,
a
number of issues are clarified.
The
high-frequency
cutoff'
of
the
harmonic radiation
depends
on the ponderomotive
energy
because the maximum
energy
of the electrons
re-
sponsible
for harmonic emission is determined
by
the
ponderomotive
energy.
Assuming
that the
ground
state is
not
significantly
depleted,
the
strength
of
the
induced
di-
pole
moment at
a
given
harmonic
depends
on
the
proba-
bility
of an
electron in
the
appropriate
velocity
range
passing
the ion.
Consequently,
it
depends
on the
ioniza-
tion
rate. The
dipole
moment also
depends
on
the
trans-
verse
spread
of the electron wave function.
Assuming
a
constant rate
of increase of the
radial dimension of the
electron wave
function,
the
strength
of the
single
atom
response
should
vary
quadratically
with
the inverse of the
laser
period.
Clearly,
harmonics will
be
generated
most
efticiently
with
the shortest
pulses
and the shortest
wave-
lengths.
The
phase
of the harmonic emission can be
es-
timated within the
quasistatic
model
by
following
the
ac-
cumulated
phase
of the free electron
over
the classical
path,
0.
5f(p/6)dx.
Phase
issues,
however,
will
be
dis-
cussed
in another
paper
[15].
Finally,
from
a
plasma
perspective,
the
plasma
collision
frequency
evolves
transiently
from
a
high
value to the
equilibrium
plasma
value.
Thus, aspects
of
high
density
Photon
Energy
(eV]
FIG.
4.
Calculated
value
of the
square
of
the
dipole
moment
(measured
in
atomic
units)
plotted
as
a
function of
Ft,
of the
harmonic.
The
calculation
was
performed for 1.06
pm
funda-
mental
radiation
interacting with
helium
with an
intensity
of
6X
10'
Wjcm2.
The
parameters
were
chosen
to match those
in
Ref.
[6].
plasma
physics
will
be
found in
low
density ultrashort
pulse
laser
produced
plasma
experiments. This
extends
the
range
of control
of
plasma
parameters
available
using
ultrashort
pulse
multiphoton
ionization
[1]. However,
only
a
slight
ellipticity
of
the
laser
polarization will
en-
sure that
the
electron never
returns
to the
environment
of
the
ion
—
the
conditions
discussed
previously
[I]
are
recovered.
In
conclusion,
the transverse
spread
of the
electron
wave
function
is an
important
parameter
which
can
be
in-
ferred
from
experiments
using elliptically
polarized
light.
Such
experiments
and their
consequences
will
form an
important new
direction
in strong-field
atomic
physics.
Future
experiments
using
linearly
polarized
light
to
study
harmonic
generation,
double
ionization, or
above-
threshold
ionization will
be of
most value
if the
polariza-
tion
of the
laser
pulse
is known
precisely.
This
paper
has
benefited
from invaluable
discussions
with
many
colleagues. These include
M.
Ivanov,
N. H.
Burnett,
P.
Dietrich, A.
Zavriyev,
A.
Stolow,
D.
Vil-
leneuve,
M.
Perry,
K.
Kulander,
and L.
DiMauro.
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New
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