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PHYSICAL REVIEW A 84, 023804 (2011)


According to the Eq. (1), we have |b1 (pst ,tst )|2 t =
b12 (pst ,tst , t)/ t = 14 EX2 dx2 [pst − A(tst )] t. From the
quantum mechanical point of view, the instantaneous transition
rate b12 (pst ,tst , t)/ t is equal to nc (W )dW , the number of
photoelectrons generated at tst (≡ t) in a small time interval
t(≡ dt) and in an energy interval dW . Because of the xuv
pulse intensity IX (t) = EX2 (t)/2, the number of xuv photons
is dNX (t) = IX (t)dt ≡ IX0 f (t)dt. With these parameters,
we have nc (W )dW = IX0 {|dX [pst − A(tst )]|2 /2}f (t)dt. If
we consider the nc (W )dW as the number of measured
photoelectrons, the right-hand side of this equation must be
multiplied by three factors: the detection efficiency η, the
geometrical factor g, and the gas atomic density ρ. Finally,
we can establish a transfer equation including the nc (W ) and
the pulse profile f (t) as follows:
f (t) = μ

nc (W ),


where μ = 2/gηρIX0 |dX [pst − A(tst )]|2 is a constant
value ([pst − AL (tst )]2 /2 ≡ ωX − Ip ). dW/dt can be
calculated with the stationary
√ point equation, W =
W0 + 2UP F 2 (t)sin2 (ωL t + ) + 8UP W0 F (t)sin(ωL t + ),
where W is the final energy of a photoelectron born at
t = tst and moving at θ = 0◦ , and Up = I /2ωL2 is the laser
ponderomotive potential. On the other hand, this equation can
be equivalently written as follows:

F (t)sin(ωL t + ) = ( W − ωX − Ip )/ 2UP .


Equation (3) is used to determine the photoelectron laser phase
ωL t or the birth time t from the measured photoelectron energy
W and to reconstruct the xuv profile f (t) for a narrow-band
xuv pulse from nc (W ). However, we cannot measure nc (W )
due to quantum interferences between electron wave packets.
Using n(W ) (that can be measured) instead of nc (W ) results
in time and photon energy errors in the PR. On the other
hand, a small energy shift ω of an xuv photon will result
in an energy shift ω of the initial energy W0 and an energy
shift W of the final energy W . This leads to a temporal
uncertainty of the determined photoelectron birth time t,
where W0 is used as a constant (central pulse energy). Using
the pulse bandwidth ωBW instead of ω, the value can be
calculated as follows: = ωBW /|dW/dt|. Theoretically,
we can calculate three factors, α [root-mean-square (rms)
intensity difference], β (mean time difference), and γ (rms
time difference), to evaluate the result of a PR in comparison
to its initial profile for a monochromatic attosecond xuv pulse.
α and γ describe the shape proximity of a retrieved pulse.
Obviously, the smaller the values of α, β, and γ , the better the
pulse reconstruction will be.

Figure 3(a) shows PRs (dotted lines) for three independent monochromatic xuv pulses (solid lines, τX = 0.250 fs,

FIG. 3. (a) PRs (dotted lines overlapped by solid lines) for
three independent narrow-band xuv pulses (solid lines centered
at td =−0.2 fs, 0, and +0.2 fs, respectively) with ωX = 90 eV,
τX = 0.250 fs, and I = 7 × 1013 W/cm2 . The error-bar lengths ( )
are calculated with ωBW = 5 eV. (b) PR (dotted line) for a three-peak
xuv pulse (solid line), each peak of τX = 0.150 fs spanned by 0.300 fs,
with ωX = 90 eV, and I = 7 × 1013 W/cm2 . (c) Similar to (b),
I = 12 × 1013 W/cm2 . (d)–(f) Similar to (a)–(c), ωX = 283.7 eV.
n(W )dW
(g)–(i) Similar to (a)–(c), ωX = 532.1 eV. (j) y(W ) =
(normalized), calculated for a zero delay monochromatic xuv pulse
with ωX = 90 eV, τX = 0.250 fs, and two laser intensities (IL ,IH )=
(2,5) × 1013 W/cm2 . (k) PR of xuv intensity profile f (t) (dotted line)
in comparison to the initial one f0 (t) (solid line). (l) PR of xuv chirp
ω(t) (dotted line) in comparison to the initial one ω0 (t) (solid line).
(m),(n) Similar to (k),(l), PRs with corrections [f (t ), ω (t )].

centered at td(0) =−0.2 fs, 0, and +0.2 fs). The error-bar
lengths represent the time uncertainties calculated with
ωBW = 5 eV for the zero delay (td(0) = 0) xuv pulse.
The minimal value of , min = 40.32 as, occurs at
t = 0.102 437 fs. By using the first reconstructed pulse
[τX(1st) = 0.253 87 fs, td(1st) = −0.00432 fs, (α,β,γ ) = (0.019,
−3.480 as, 4.300 as)] as the input data of the PES calculation, performing the second PR yields a pulse duration
of τX(2nd) = 0.258 34 fs and td(2nd) = −0.004 18 fs. The time
differences between the first and second PRs are used to
correct the first retrieved times (t), giving a final PR of
τX = 0.250 32 fs, td = +0.16 as, td ≡ td − td(0) = +0.16
as, and (α ,β ,γ )=(0.00102, −0.113 as, 0.324 as). PRs for
pulses with td(0) = −0.2 and +0.2 fs have similar results: τX =