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Einstein centennial review article / Article de
synthèse commémoratif du centenaire de
l’année miraculeuse d’Einstein
Attosecond science and technology1
Jérôme Levesque and Paul B. Corkum
Abstract: Attosecond technology is a radical departure from all the optical (and collision)
technology that preceded it. It merges optical and collision physics. The technology opens
important problems in each area of science for study by previously unavailable methods.
Underlying attosecond technology is a strong laser field. It extracts an electron from an
atom or molecule near the crest of the field. The electron is pulled away from its parent ion,
but is driven back after the field reverses. It can then recollide with its parent ion. Since
the recolliding electron has a wavelength of about 1 Å, we can measure Angström spatial
dimensions. Since the strong time-dependent field of the light pulse directs the electron with
subcycle precision, we can control and measure attosecond phenomena.
PACS Nos.: 33.15.Mt, 33.80.Rv, 39.90.+d, 42.50.Hz, 42.65.Ky
Résumé : La technologie à l’échelle de l’attoseconde se démarque dramatiquement de
toutes les technologies optiques précédentes. Elle fusionne l’optique avec la physique des
collisions. Cette technologie ouvre, dans tous les domaines des sciences, de nouvelles portes
pour étudier des phénomènes auparavant hors de portée. À la base de la technologie de
l’attoseconde se trouve un champ laser intense. Près de la crète du signal, on peut extraire
un électron d’un atome ou d’une molécule. L’électron est retiré de son ion parent, mais y
retourne lors du renversement du champ. Il peut alors recollisionner avec l’ion. Puisque cet
électron a une longueur d’onde associée de l’ordre de 1 Å, des mesures spatiales à l’échelle
de l’Angström sont possibles. Parce que le fort champ dépendant du temps de l’impulsion
lumineuse dirige l’électron avec une précision temporelle atteignant le demi-cycle optique,
nous pouvons contrôler et mesurer à l’échelle de l’attoseconde.
[Traduit par la Rédaction]
Received 9 December 2005. Accepted 9 December 2005. Published on the NRC Research Press Web site at
http://cjp.nrc.ca/ on 31 January 2006.
J. Levesque and P.B. Corkum.2 National Research Council Canada, 100 Sussex Dr., Ottawa, ON K1A 0R6,
Canada.
1
This article is one of a series of invited papers that have been published during the year in celebration of
the World Year of Physics 2005 — WYP2005.
2
Corresponding author (e-mail: [email protected]).
Can. J. Phys. 84: 1–18 (2006)
doi: 10.1139/P05-068
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Can. J. Phys. Vol. 84, 2006
1. Introduction
Lasers were demonstrated experimentally in 1960. Thanks to the laser, we have gained the ability to
measure directly very short time intervals. Usually this is done by using a pair of very short laser pulses,
one delayed with respect to the other. One pulse excites the dynamics to be observed and the second
one probes its evolution as a function of the time delay [1]. Attosecond technology pushes ultrafast
measurement to new extremes and opens new ways to make fast measurements [2, 3] and new media to
study [4–8].
The ability to generate attosecond pulses is the culmination of 40 years of short-pulse laser science
and technology. The attosecond time scale is the natural time scale for electron motion. Thus, we have
reached the most important milestone for atoms, molecules and solids.
Figure 1 shows a time line of the duration of laser pulses. There was a steady decrease in pulse
duration from 1960 to 1986. In 1986, 6 fs pulses were achieved [9]. At three periods of the then used
600 nm light, it was clear that incremental improvements were no longer sufficient. New technology
was needed. This paper is about the new technology and the science that it implies. In essence, the ideas
behind the technology were introduced in three papers. One introducing the underlying concept [10],
one outlining how it could lead to attosecond pulses [11], and one introducing the basic idea of how
pulses could be measured [2].
There are three ways to look at attosecond technology. They go to the very heart of quantum physics.
• From a nonlinear optics perspective, attosecond pulses are generated in a highly nonlinear process. The process is so nonlinear that attosecond pulse generation cannot be understood using
perturbation theory. However, classical physics is helpful [10, 12]. We will see that the high nonlinearity allows us to access processes that are shorter than one period of the driving laser. Much
of attosecond technology exploits subperiod phenomena.
• From the perspective of electrons as particles, underlying attosecond optical pulse generation are
attosecond electron pulses (or bunches in the language of collision physics) [6]. These attosecond
electrons can be used directly in collision physics experiments [5, 13] or they can transfer their
kinetic energy and their time duration to attosecond photons [11,14]. From a particle perspective,
a nanoscale electron accelerator lies at the heart of attosecond pulse generation.
• The electron wave perspective emphasizes the wavelike nature of quantum particles. From that
point of view, attosecond technology can be understood as electron interferometry [2]. This
approach allows for many analogies with familiar concepts in optics.A great deal of the technology
and its implications can be understood in this way.
In this paper we will introduce attosecond technology mostly via interferometry, stressing the wave
nature of the electron, but we will note other perspectives as well.
1.1. Tunnel ionization: creating the beam splitter for the atomic interferometer
Figure 2 depicts the potential that an atom’s electron would experience while immersed in the light
wave (everything that we say about Figs. 2 and 3a would be much the same if it were a molecule).
The electron is confined to an atom by the Coulomb potential. A laser field exerts a time-dependent
force on the bound electron. If it becomes large enough, the electron can tunnel from the atom [15, 16].
Tunnelling splits the electron into a bound wave function and a continuum wave packet that escapes the
ion. Tunnelling is the beam-splitter in the electron interferometer that the light pulse will create from
the atoms own electrons.
Figure 3a is a one-dimensional cut through the atom in Fig. 2. As the laser light pulse passes an
atom, the electron may have multiple chances to tunnel. Thus, the electron can split many times during
a long pulse.
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Fig. 1. A time line showing the duration of the shortest optical pulses that could be produced. You will see
that we have just crossed the attosecond threshold. The Bohr orbits times for valence electrons in atoms and
molecules is ∼100 as. Pulses shorter than ∼100 as allow us to freeze electron motion.
Fig. 2. A sketch of the influence of the laser electric field on the potential confining an electron to an atom.
(a) Unperturbed Coulomb potential. (b) Addition of the laser field to the Coulomb potential. There is an
analogy with tipping a cup of water.
1.2. Wave packet motion: creating a delay line for electrons
Once the electron is split, the attraction of the negative electron to the positive ion rapidly decreases.
In contrast, the force that the light exerts on the electron does not depend on how close the electron
is to the ion. Therefore, the force exerted by the light wave controls the electron wave packet motion.
At first the electric field of the light pushes the wave packet away from the ion, but soon the electric
field reverses direction and the force begins to push the electron back [10, 12]. Parts of the wave packet
return and pass over their first point of origin. In Fig. 3a, the motion of the electron is represented by
the red arrow. Figure 3a shows the wave packet first tunneling out of the potential well (bottom wave)
and, about half an optical cycle later, sweeping over the ion (top wave).
Figure 3b emphasizes the deep analogy between the recollision electron and an optical interferometer. On the left is the electron interferometer, and on the right, the optical interferometer. We have made
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Fig. 3. (a) A one-dimensional cut through the potential in Fig. 2. In this simpler picture is included the
electron wave function. It has two parts: the bound part, trapped by the ion, and the wave packet that is
pulled free of the ion. (b) An optical interferometer is compared with the laser induced electron recollision.
(a)
(b)
a beam splitter for an electron by ionizing the electron with an intense laser pulse. We have found a
method to redirect the electron back to where it came from. There, it must interfere with its former self
just like any other wave interferes. This process is often termed a “recollision”. The term recollision
emphasizes the particle-like nature of electrons while the electron interfering with itself emphasizes the
wave-like nature of the electron.
1.3. Electron–ion recollision: creating an electronic interferogram
Collisions involving energetic particles — particles with kinetic energy ≥10 eV — occur in much
less than one femtosecond. Thus, many phenomena in collision physics are attosecond phenomena.
Attosecond science emerges from these fast collisions. However, although collision experiments have
worked in the attosecond time scale for decades, it was very hard to make use of this time scale for
measuring fast dynamics since collisions cannot be timed with respect to another event.
Thus, recollision can have a major impact on collision physics. It is the laser field that times the
collision. Since laser pulses can be easily synchronized and delayed, the collision can be synchronized
to one or more optical pulses. The synchronization can be of attosecond precision.
Recollision will make collision experiments more and more like optics experiments. Together they
will open new types of experiments. To emphasize the symbiosis, we now discuss how collision techniques can be used to determine the characteristics of the recollision electron.
1.3.1. The probability of recollision
In quantum mechanics, the way measurements are made determines what we can learn about the
object that we measure. We want to determine at what time and with what probability an electron
that ionizes recollides. We use a hydrogen molecule for the measurement, together with an 800 nm
wavelength and a 50 fs laser pulse [6]. Ionization may occur over many half cycles of the pulse. Although
we do not know exactly if or when the electron makes it through the beam-splitter (i.e., ionizes), the
molecule itself can tell us a great deal. If we measure a hydrogen molecular ion for example, we then
know that the molecule ionized. Furthermore, whenever the electron is removed from the hydrogen
molecule, one of the two electrons binding hydrogen is removed. As the electron departs, the bond
weakens and the molecular ion stretches. That is, it starts to vibrate and we use this vibration as a clock
to tell the time between ionization and recollision.
If the electron that has been removed recollides with the ion, it can dislodge the one remaining
electron. Should this occur, the vibration ceases and the molecule is transformed into an excited hydrogen
molecular ion or two hydrogen ions in close proximity. These channels can be distinguished by correlated
ion measurements [17]. We determine the probability of recollision by comparing the number of atomic
hydrogen ions that are produced to the total number of ionization events.
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Fig. 4. The probability of recollision shown as a function of time following ionization. The laser electric
field is plotted for convenience. The electron is most likely to recollide about 1.7 fs after it detaches.
For simplicity, we describe the case of double ionization [17], but the first study used excited
molecules [6, 18]. Since we know that like charges repel and that the kinetic energy they gain depends
on how close they were to each other when the second electron was dislodged, measuring the ions kinetic
energy reveals how far they were separated when the recollision occurred (in other words, measuring
the delay between ionization and recollision).
Figure 4 shows the time dependence of the probability of recollision (plotted as a current density).
The reader is referred to ref. 6 for details. In brief, Fig. 4 is obtained by following the classical trajectories
that are populated by tunneling. Figure 4 can then be used to simulate the results of the experiment.
Agreement is assumed to yield confirmation of the calculated time-dependent probability of recollision.
This is, in essence, a collision physics experiment. Therefore, instead of plotting the probability
of recollision, we adopted the language of collision physics. That is, we plot the current density of
the recollision electron. This represents the current per unit area of an electron beam that is needed to
produce the same probability of double ionization and the same time dependence. The current densities
represented by the recollision electron are much beyond current densities that can be produced by
laboratory sources and, of course, their time scale is unique. The electron is most likely to return about
1.7 fs after it has left, and most of the probability of the electron recolliding is confined to about 1 fs
(1000 as).
Not shown in Fig. 4 is the fact that those electrons that recollide early in the pulse (for example
about 1 fs after ionization) have low velocity (long wavelength). Those that recollide near the peak of
the electron pulse (about 1.7 fs) have high velocity (shorter wavelengths). Those that recollide after the
pulse peak (for example about 2.5 fs) have longer wavelengths again. It is this well-confined recollision,
combined with the energy sweep that accompanies it, that allows us to produce attosecond pulses.
If the electron misses the first time, it has another chance following the next field reversal. The
subsequent electron pulses in Fig. 4 are separated by 1/2 laser period and represent repeated attempts
at recollision driven by the oscillating field.
1.3.2. Recording the electron interference
We have now shown that, when intense laser pulses ionize atoms or molecules, an electron interferometer is naturally formed. The interference modulation is deep since the probability of recollision is
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Fig. 5. A three-dimensional depiction of the time-dependent electronic interference. In this example, a
1s orbital is superposed coherently to an electronic plane wave. (a) The real part of the superposition is
shown. (b) The squared modulus of the superposition. In both (a) and (b), the bottom picture represents the
evolution of the superposition after a time corresponding to a displacement of the plane wave by pi radians
from (a).
large. Since the electron wave length is ∼1Å, the interference structure is, therefore, on the Angström
scale. But if we are to use this electronic interferometer, we must find a way to “see” the interference.
Figure 5 (which is a two-dimensional version of Fig. 3a) concentrates on the region where the
recollision electron and the initial wave function overlap. This is where the interference actually occurs.
In Fig. 5a, the blue wave in the top and bottom images represents the recollision electron. On the scale
of this illustration, the recollision electron looks like a uniform wave passing over the initial bound
state. The left and right images represent the wave at two closely spaced instants. The recollision wave
has advanced by 1/2 wavelength between those images.
When waves interfere, we add the overlapping components of the wave — in this case, the recollision
wave packet and the bound-state wave function. The biggest peak on the sum wave moves from one
side of the atom to the other between the two images along the direction of the laser field. (We will use
this motion to determine the structure of the bound state wave function later in the paper.) We highlight
amplitude of the superposition in the colour coding.
The square of the wave functions, shown in Fig. 5b, represents the probability of finding the electron
in any region of space. The additional projections at the bottom of each image in Fig. 5b are included
as an alternative (and perhaps easier) way to visualize the process.
Figure 5b shows that interference between the bound and the recollision electron transfers the
charge from left to right. This rapid oscillating motion continues as long as the recollision lasts. The
faster the recollision electron moves, the faster the oscillation. An oscillating dipole emits radiation at
the frequency of the oscillation. Thus, we have a method of reading the electron interferometer and
simultaneously a source of short wave length light.
2. The strong field approximation
So far our description of attosecond science has been through physical images, not mathematical
equations. Mathematically speaking, the radiation emitted during recollision is actually given by the
dipole acceleration, i.e., the second derivative of x(t) = |r|, where x(t) is the time-dependent
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dipole moment, | is the electronic state and r is the position operator. The electronic state | can be
expanded into a superposition of two components, a bound part |bound and a continuum wave packet
|cont , which are given by
|bound α(t) · |ψ
+∞
d3 k β(k, t) · |k
|cont =
−∞
(1)
(2)
In (1), |ψ is the stationary bound state. In (2), each |k represents the state of an electron in
the continuum, with momentum k. As already mentioned, this free-bound duality in a single electron
comes from the tunneling ionization process, which acts as an electronic beam splitter. We can write an
expression for the dipole in terms of a superposition of these states
x(t) = bound |r|bound + cont |r|cont + cont |r|bound + bound |r|cont (3)
The first term represents the static dipole moment of the atom or molecule. The second term is
continuum–continuum radiation — called bremhstrahlung in plasma physics. The two last terms describe, respectively, the detachment of an electron and the electron recombination to the ion core. In
the following, we will consider only the recombination term, bound |r|cont , which is responsible for
high-order harmonic emission and attosecond pulse generation [19].
The strong-field approximation assumes that the electron wave packet in the continuum moves like a
free particle with no influence from the ion. This approximation is justified by the fact that, for electrons
with sufficient kinetic energies, the interaction with the ion core becomes negligible. Consequently, the
continuum states can be approximated by plane waves exp[i(k · r + ωe t + φ(k, ωe ))], where ωe is the
de Broglie frequency associated with a travelling electron of momentum k. Assuming, furthermore,
that there is negligible depletion of the ground state (i.e., α(t) 1 in (1) one obtains the following
expression for x(t), the time-dependent dipole:
+∞
+∞
dr ψ ∗ · r ·
dk β(k, t) × exp i k · r + |k|2 t + φ(k)
x(t) −∞
+∞
−∞
dk β(k, t)
−∞
+∞
−∞
dr ψ ∗ × r × exp i k · r + |k|2 t + φ(k)
(4)
where we used the fact that the electronic de Broglie frequency, ωe , is given by ωe = me (p · k) =
m2e |k|2 = |k|2 (in atomic units, me = = 1).
In the semiclassical model developed by Lewenstein et al. [19], the Keldysh approximation [15] is
used to evaluate the time-dependent β(k, t). The expression for the dipole is then given as a function
of the canonical momentum p and the electromagnetic vector potential A
+∞
t
dt dp ψ|r|p + A(t) × exp −iS(p, t, t )
(5a)
x(t) = −i
−∞
0
× E(t ) · p + A(t )|r|ψ
(5b)
where
S(p, t, t ) =
t
t
dt
2
p + A(t )
+ Ip
2
(6)
This integral can be associated to the following mechanism: after the electron has left the |ψ state by
a dipole transition at t (term 5b), it evolves freely in the laser field from t to t, and then recombines to
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|ψ at time t by another dipole transition (term 5a), generating harmonic radiation. Within that model
and considering an oscillating field E(t) = E cos(ωL t), β is, therefore, given by [19]
t
β(k, t) =
dt E cos(ωL t ) · p + A(t )|r|ψ
(7)
0
Equation 4 can also be written in a simpler form if we use the fact that, for electrons with high energies,
β(k, t) varies slowly compared to exp[i|k|2 t] , and thus does not contribute to the high-frequency
oscillation of the dipole. For these high energies, we can also neglect the electron trajectories that are
not along the laser driving force at the moment of recollision. We thus exclude the transverse momenta
components (y and z) as well as electrons with negative momenta. Moreover, the classical model, which
is valid in the strong field approximation (SFA), associates a unique recollision momentum k for each
time τ . We can thus simplify (4) as
+∞
dr ψ ∗ × r × exp i k(t)x + k(t)2 t + φ (t)
(8)
x(t) β (t)
−∞
where k(t) and φ(t) are the recollision momentum and phase determined by the classical trajectories
for each time t. The amplitude of the dipole oscillation, in the frequency domain, is then
x(ω) Acont (ω) × Abound (ω)
(9)
where
√ Acont (ω) = β (τ ) exp iφ( ω)
+∞
√
dr ψ ∗ × r × exp[i ωx]
Abound (ω) =
−∞
(10)
(11)
where we used the fact that the frequency of the harmonics, ω, is the same as the electronic frequency,
ωe (and that ωe = k 2 ). Within our model, the high harmonics spectrum can thus be interpreted as the
product of two complex amplitudes, Acont and Abound , each one taking into account the participation
of the continuum state and the bound state, respectively in the rapid dipole oscillation producing the
high harmonics. Both of the amplitudes are complex, and the vector nature of Abound determines the
polarization state of the harmonics. Knowing ψ, Abound can be calculated. If we can also determine Acont
the single atom nonlinear response, x(t), is then known . Alternatively, if we measure x(ωH )/Acont , we
can determine ψ. This idea can be applied for orbital imaging and is covered in Sect. 3.
2.1. High-harmonics generation: attosecond pulse trains
Electron recollision, by producing an oscillating charge through its interference with the groundstate wave function, produces radiation at the frequency of the oscillation. Observing this light with
a spectrometer, we read the interference pattern. The duration of this emission is determined by the
duration of the electron recollision, The frequency of emission sweeps from low to high frequency and
back again just as the electron energy sweeps.
If the driving laser field is multicycle, then a new electron wave packet is created each 1/2 period
and the atom experiences a sequence of recollisions. When Fourier transformed, this periodic emission
appears as a ladder of harmonics extending to the maximum kinetic energy (KE) of the recollision
electron plus the ionization potential of the atom Imax = KE + I P [19]. The shortest wavelength
light that has been produced in this manner is 1.2 keV [20]. The maximum kinetic energy is adequately
estimated by classical physics [10]. In a real experiment, shown schematically in Figs. 6 and 7, the
light pulse interacts with a gas of atoms. As long as the density does not become too high and the
interaction region is confined, then the emission from each atom in the volume adds coherently. That
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Fig. 6. A sketch of the optical parts of the experimental set-up for the tomography measurements. The
initial pulse is first split in two copies by a beam-splitter (BS1). The pulse transmitted through B1 is to be
used as the alignment pulse, and is sent in a variable delay line. The reflected pulse is used to generate
harmonics. Before the two pulses are recombined on BS2, the polarization angle of the alignment pulse is
controlled by a zero-order half-wave plate, to scan the molecular alignement angle with respect to the axis
of polarization of the ionizing pulse.
Fig. 7. Schematic of the vacuum set-up used for producing high harmonics. The harmonics are produced
by focusing intense IR pulses ( 1014 W/cm2 at the focal point) in a supersonic gas jet. The harmonics are
then dispersed by a concave XUV grating (1200 lines/mm nominal) with a variable groove separation. The
grating focuses each order on the same plane, in which we put a MCP detector. There is a phosphor screen
at the back of the MCP. It is filmed by a CCD camera and the image is recorded on a computer.
is, high harmonic generation is phase matched. Phase matching filters out the late recollisions shown in
Fig. 4. However, many other characteristics of the harmonic output are forged at the single atom level.
Now we turn our attention to controlling the recollision electron. It is the potential for control [11,21]
that makes the recollision electron such a powerful concept for attosecond science. Once the electron
wave packet is formed, the influence of the ion recedes and we exercise control through the laser field.
Single attosecond pulses are generated by controlling the recollision so that it can only occur once.
Thus, attosecond pulse generation reduces to gaining a high degree of control over light pulses, which
is then transferred to controlling recollision electron wave packets.
2.2. Carrier envelope phase stabilization
As we have seen, attosecond science is subcycle science. For the first time in optics, it is insufficient
to control only the pulse envelope. The time-dependent field of the laser pulse must be controlled. In
other words, both the envelope of the pulse and the field oscillation within the envelope must also
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Fig. 8. Illustration of carrier envelope phase. The pale and dark traces show two different carrier envelope
phases for IR pulses (λ = 0.8µm) of the same duration. The pale trace is an even ("cosine") pulse, while
the dark trace is an odd (“sine”) pulse. Superposed to the two IR pulses is the (idealized) attosecond pulse
which would be obtained with an IR cosine pulse.
be controlled. Figure 8 shows the time-dependent field for two pulses with identical envelopes. Only
the phase of the electric field oscillation within the envelope, known as the carrier envelope phase (or
absolute phase) is different for these two pulses. The carrier envelope phase is not a firm characteristic
of a pulse. For example, as the pulse propagates, the carrier envelope phase evolves because the group
velocity and phase velocity are different in most media.
Until the last few years, there was little thought given to controlling the carrier envelope phase. It
is not important in linear optics and did not seem important for low-order nonlinear processes excited
with multicycle pulses. However, the classical approximation that we have just discussed shows that
the carrier envelope phase will be important for all strong field (highly mulitphoton) processes. In fact,
for few cycle pulses where the pulse bandwidth can be so large that the pulse interferes with its second
harmonic, the carrier envelope phase becomes important even for second-order processes.
The technology of producing phase-stabilized pulses from oscillators was recently developed for
optical standards [22]. We will not review these developments here, but refer the reader to a recent
review, ref. 23. With careful design these pulses can be amplified and compressed, producing energetic
phase-stable few period pulses [24]. These pulses are essential for stably producing single attosecond
optical pulses.
2.3. Single attosecond pulses
Once the carrier envelope phase of few cycle pulses can be controlled, there are many ways to
produce single attosecond pulses. The first method proposed [11] was to use a pulse with time-dependent
polarization. The pulse begins circularly polarized, moves rapidly through linear polarization, and ends
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circular. If you follow the classical trajectories of electrons emitted at different times in the pulse you
will find that there is only a brief interval near linear polarization during which the electron can recollide.
An alternative way of understanding this is that, in a highly multiphoton process where many photons
are transformed into a single high-frequency photon, the number of right and left circularly polarized
photons must nearly balance for emission to be possible. This is a highly restrictive condition that is
only achieved for a brief interval during the pulse. Attosecond pulses have just been achieved in this
manner3 .
However, the first isolated attosecond pulses were produced differently. If you follow the classical
trajectories of electrons emitted at different times during a 6 fs linearly polarized pulse, you will find
that only electrons formed during a small part of one of the laser periods during the pulse rise-time can
recollide with maximum velocity. At all other times the recollision energy is lower. That means that,
if we observe only the shortest wavelength radiation, by adding a cut-off filter to the set-up shown in
Fig. 7, the radiation that is observed can only be emitted over a very short time interval; 250 attosecond
pulses are currently achieved [25] following this procedure, the shortest duration pulses plotted in Fig. 1.
Forming attosecond pulses with time-dependent polarization promises much larger bandwidth pulses
and, therefore, the potential for measuring much faster processes. In fact, it is possible to use chirped
pulses for single electron wave-packet measurements (and perhaps many other type of attosecond
measurement) as effectively as transform limited pulses of the same bandwidth [26]4 . It seems likely
that the limit to attosecond measurements will soon reach lower than 25 as, the atomic unit of time. But
how can attosecond optical pulses be measured?
2.4. Measuring the duration of attosecond pulses: the attosecond streak camera
We measure femtosecond optical pulses by making a replica optical pulse and the two pulses measure
each other through a low-order nonlinear optical processes in selected materials. This is most easily
seen via autocorrelation. The two pulses are combined in a nonlinear crystal phase matched for secondharmonic generation. If the two pulses arrive together in the crystal, then the second harmonic is more
effectively produced than if the pulses arrive separately in time. The second-harmonic signal measures
the overlap of the pulses, and hence their pulse duration.
This procedure is difficult to extend to attosecond XUV pulses. One important question is what can
one use for beam splitters and what is an effective nonlinear medium for XUV light. Autocorrelation
has only recently been demonstrated and only for the longest wavelength XUV radiation [27, 28]. But
there is a very attractive alternative.
Instead of measuring the attosecond pulse directly, it is equivalent to measuring a photoelectron
replica of the attosecond pulse [2, 21, 29]. A photoelectron replica can be easily produced by photoionization. Photoionization serves as a photocathode in our attosecond streak camera [21, 30]. If the
photoelectrons are produced in a strong laser field, then the field labels the time of birth of each component of the photon wave packet just as it controls the photoelectron wave packet produced by tunnel
ionization to produce attosecond pulses. The fundamental field that produced the attosecond pulse is
the deflection field in our attosecond streak camera. By measuring the energy and direction of the photoelectron, we observe the influence of the streaking field. Knowing the field, we measure the pulse
duration. Thus, attosecond measurement can be achieved in almost direct analogy to an optical streak
camera [2, 14].
Thus, the critical advance that allowed attosecond measurement is the concept of producing a
photoelectron replica of the pulse and measuring it. Once this is understood, it is possible to find
attosecond analogues for SPIDER [?] and FROG [2, 25]. There is currently no lower bound to our
ability to measure attosecond pulses, at least in principle.
3
4
Dr. É. Constant. Private communication. 2005.
G.L. Yudin, A.D. Bandrauk, and P.B. Corkum. Phys. Rev. Lett. Manuscript in preparation.
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Fig. 9. Ratios of momentum distributions for an electron wave packet diffracted by a diatomic core. (a)
Diffraction of a Gaussian wav epacket with its initial momentum matching the effective cutoff momentum in
a recollision (I = 3 × 1014 W/cm2 , λ = 400 nm). (b) Laser-induced diffraction at the same intensity. (Figure
taken from [32]).
(a)
(b)
2.5. Implications of attosecond technology for collision physics
We have emphasized that many aspects of attosecond science are related to collision physics. In fact,
recollision is a controlled collision, control being exercised by the laser field. Recollision allows us to
time the collision precisely with respect to an optical pulse. In this way, much of the power of optics is
transferred to collision physics. One example of how the timing precision can be used is measurement of
the motion of a vibrational wave packet in the D+
2 molecule [6]. As we have discussed above (Sect. 1.3.1),
tunnel ionization starts a vibrational wave packet on the ground state of D+
2 . Recollision, timed by the
field, stops the wave packet motion, transferring the remaining electron to an excited state or to the
double ion. Knowing the excited state potential surface, the fragment kinetic energy determines the
internuclear separation at the time of recollision. Thus, we can follow the wave packet motion with
attosecond timing precision by using different wavelength light (800 nm to 1.8 µm) to control the
recollision. Reported in ref. 3, this is the highest time resolution measurement achieved so far with
attosecond technology.
There are many ways that recollision can open new opportunities for collision physics experiments.
For example, recollision might be controlled by a probe pulse in a pump-probe measurement. Then
the recollision electron is a probe electron pulse. Since the recollision electron has a wavelength of
∼1Åand is delivered to an atom or molecule with very high timing precision and high current density,
recollision may allow diffraction experiments to be very efficiently carried out on molecules, even
those undergoing dynamics. Known as laser-induced electron diffraction, this process was proposed in
1996 [31]. Figure 9b shows a calculated diffraction pattern found by solving the three-dimensional timedependent Schrödinger equation for a two-atom, one-electron molecule [32] with its internuclear axis
aligned perpendicular to the laser field. Plotted is the ratio of the electron probability scattered parallel
to the internuclear axis to that perpendicular to the axis. The diffraction pattern is clearly evident in the
image. Figure 9a is for an external, monoenergetic electron source. It is included for reference.
Collision experiments are often concerned with the physics of electron–electron interactions in
atoms or molecules. Electron–electron interactions are equally important for recollision. However, for
recollision we can use laser methods to excite the molecule or to align it. That is, the electronic state, or the
scattering potential is determined at the time of collision. If decay products from a collision emerge while
the laser pulse is still present, then the strong laser field streaks any charged particle, just as photoelectrons
© 2006 NRC Canada
Levesque and Corkum
13
are streaked by the field. The time scale for some of the elementary collision processes is only now
being measured [5, 13]. Streaking, important at intensities of ∼ 1014 to 1015 W/cm2 will become even
more so at higher intensities. By intensities of 1022 W/cm2 (just now becoming experimentally feasible)
even nuclear decay can be stimulated by recollision and timed by streaking [7, 8].
We have now reviewed the physics underlying attosecond technology, discussed how attosecond
pulses are produced, and how they are measured. Throughout we have emphasized the close connection
between attosecond science and collision physics. In the process, we discussed how recollision electrons
will allow us to determine the position of atoms in a molecule by laser-induced electron diffraction.
Thus, attosecond science offers the potential for Angström scale measurements. Attosecond science will
actually be Attosecond–Angström science. We conclude by returning to interferometry and attosecond
pulse generation, showing how attosecond pulse generation is connected to electron imaging [33].
3. Tomographic imaging of molecular orbitals
The integral in (8) shows how the rapid dipole oscillations produced at the moment of recombination
depend directly on the quantum state of the electron. Within our model, that state is represented as the
coherent superposition of a stationary bound state and a free state, corresponding to the electron travelling
in the continuum. As already mentioned, the origin of this free-bound duality in a single electron comes
from the tunneling ionization process, which acts as an “electronic beam splitter”. The high-harmonic
spectrum thus contains information about the free-bound electronic superposition, and can actually be
seen as a measurement of the recombining electron’s quantum state.
Equations (8) to (11) show how the high-harmonic spectrum amplitude depends both on the strongfield ionization rate of molecules (the Acont term) and on the dipole moment for the transition from the
continuum back to the bound state (the Abound term). Since the ionization dynamics in the tunneling
regime depend mostly on the ionization potential, Ip , it is possible to determine Acont by combining
the experimental measurement of x(t) (i.e., the harmonic spectrum), with a calculation of Abound for a
reference atom. That is, we can use a well-known atom as a reference for a molecule with a similar Ip .
We can then use that semi-empirical Acont to normalize molecular spectra, and thus obtain the Abound
amplitude for molecules.
To discriminate different alignment angles, we align molecules using femtosecond laser pulses that
generate rotational wave packets [34] and probe the ensemble at the time of maximum alignment. We
rotate the polarization of the probe pulse in 5◦ intervals, from 0◦ to 90◦ , to obtain 19 harmonic spectra.
The experimental set-up is shown in Figs. 6 and 7.
In Fig. 6, we show the optical set-up used to split the initial laser pulse (800 nm, 25 fs, 6 mJ) in
two copies: an alignment pulse (0.8 mJ, 50 fs) and a probe pulse (2.2 mJ, 25 fs), used to produce the
harmonics. Both pulses are linearly polarized. The alignment pulse stretches naturally to 50 fs as it
propagates through the beam splitters (BS1 and BS2) and the half-wave plate. The alignment pulse is
first advanced with respect to the probe pulse by a computer-controlled delay stage. Before the two
pulses are recombined, the polarization of the alignment pulse is also rotated by a motorized zero-order
half-wave plate, to determine the molecular alignment angle. The two pulses are then focalised into a
supersonic gas jet, where the molecules are aligned and the harmonics are generated. The harmonic
beam propagates in vacuum to an XUV spectrometer, where the amplitude of harmonics are recorded.
Looking back at (11) for the analytic formulation of Abound , we see that it actually corresponds to
the one-dimensional spatial Fourier transform of the three-dimensional bound state projected onto the
polarization axis of the laser (x)
+∞
√
Abound (ω) =
dr ψ ∗ × bmr × exp[i ωx]
−∞
+∞
=F
dy dz ψ ∗ r
(12)
−∞
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Can. J. Phys. Vol. 84, 2006
Fig. 10. Tomographic imaging of the highest occupied orbital in N2 . (a) shows the experimental results used
for the reconstruction (which are obtained in k-space), (b) the reconstructed highest occupied orbital in N2 ,
and (c) calculated highest occupied orbital in N2 (3σg orbital, projected in a plane). The horizontal axis in
(b) and (c) corresponds to the internuclear direction.
A high-harmonic spectrum, once normalized with respect to Acont , thus provides a measurement of
Abound , which contains information about a projection of the bound state. By recording many projections,
we recover the probability amplitudes of the bound state itself, in momentum space. The recovery of an
object from such a set of projections is possible through the use of the Fourier Slice Theorem, which is
at the base of most tomographic methods [35].
The high-harmonic spectra thus provides a measurement of the probability amplitude of the electronic bound state, in momentum space. At this point, phase information is missing to perform an
inverse Fourier transform and obtain an image in position space. One solution would be to improve
the experimental set-up, so that a phase measurement can be done directly on the harmonics. Such
phase measurements have been demonstrated experimentally [36]. Another possibility is to make an
assumption on the phase, based on the knowledge we already have about the molecule and on arguments
regarding the continuity of the wave function.
In an attempt to recover an image of the bound electronic state of N2 , we recorded the spectra for
different alignments of the molecule. We normalized that set of spectra with respect to argon, in the
manner described above (argon and N2 have similar Ip’s, 15.8 and 15.6 eV, respectively), to obtain
an experimental measurement of |Abound |2 for N2 . That measurement is shown in Fig. 10a. We then
took into account the homonuclear nature of N2 to assume that the electronic bound state must be
either strictly even or odd. Also, for the points at which the normalized spectra reached zero amplitude
(these points form a dark ring in Fig. 10a), we assumed a π -phase shift in the wave function. These
two assumptions were enough to estimate the phase of the electronic state in momentum space. An
inverse Fourier transform was then possible, using the normalized power spectrum in combination to
the assumed phase. The resulting image, obtained assuming an even wave function, is shown in Fig. 10b,
along with an image of the highest occupied molecular orbital of N2 in Fig. 10c.
There is a close association between the molecular tomography method that we outlined here
and the more familiar application of tomographic imaging in medicine. When a patient goes to the
hospital for a tomographic scan, a series of two-dimensional X-ray images are taken, with the X-rays
passing at different angles through the patient’s body. Each of these images is a projection. In our
experiment, the patient is replaced by a molecule, and the X-rays by recombining electrons. One subtle
but important difference though is that in medical imaging, the projection itself is recorded, and Fouriertransformed later by a computer. In the molecular case, as (12) shows, the Fourier transform is provided
by the physical process itself. While in hospitals the spatial absorption profile that is recorded is a real,
positive function, in the molecular case the full measurement involves a measurement of amplitude and
phase in the spectral domain. Spectrometers for XUV light have been around for decades, allowing
the measurement of spectral amplitudes. The measurement of XUV spectral phase remains technically
challenging though. Nevertheless, harmonic phase measurement has recently been demonstrated by a
© 2006 NRC Canada
Levesque and Corkum
15
Fig. 11. A sketch of the idea behind attosecond electron dynamics measurements. Although the initial
electronic state can be a coherent superposition of many eigenstates (here an s and a p atomic orbitals), the
tunneling electron comes almost exlusively from one of these eigenstates; the one that extends further away
from the center of the atom (i.e., the p orbital). On return, nevertheless, it interacts with the whole volume
of the atom, and both s and p orbitals are involved in the interaction.
group at CEA (France) for the total phase of harmonics [36] and in our group for the phase of Acont 5 .
The stage is thus set for a full implementation of tomographic methods to the atomic and molecular
domains.
4. Attosecond imaging — seeing electrons move
When a molecule’s atoms move during a chemical reaction, they carry their electrons with them.
Our technology is fast enough to image this motion. This would fulfill the dream of truly imaging the
electronic changes that occur during a photo-chemical reaction. As we extend the technology to more
complex molecules and molecules undergoing dynamics, we will obtain images of chemical reactionsin-progress that will be as dramatic as Muybridge images of trotting horses, taken more than 125 years
ago [37].
But attosecond technology promises much more — it is capable of attosecond time resolution. Of
course, for attosecond imaging to be worthwhile, there must be interesting things happening on this
time scale to observe. Attoseconds are the time scale for electron motion. A classical electron completes
a Bohr orbit in about 150 as. Although quantum mechanics says that the Bohr orbit is not observable,
it does allow us to observe something comparable. We now show one way to measure this bound-state
electron wave packet — with attosecond precision.
Figure 11 illustrates the idea [26]. This time we split the electron twice. First, we split the electron
to make the bound-state wave packet. Such wave packets require population in two or more orbits.
These interfere with each other just as our recollision electron interferes with its parent orbital, with
their interference corresponding to the attosecond motion that we wish to measure. (This first splitting
can occur naturally during the process that we want to study, or we can stimulate its occurrence with
resonant light.)
We then split the electron a second time. This time, we use an intense light pulse just as we described
in the earlier sections of the paper. Almost certainly it is the upper state that ionizes, since it is less
strongly held by the ion. When the electron wave packet recollides, it interferes with both bound parts
5
N. Dudovich, O. Smirnova, J. Levesque, M. Ivanov, D.M. Villeneuve, and P.B. Corkum. Manuscript in preparation.
© 2006 NRC Canada
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Can. J. Phys. Vol. 84, 2006
Fig. 12. Shows the calculated high-harmonic spectra from a bound-state electron wave packet with a period
of (a) 290 as, (b) 330 as, and (c) 444 as. The initial wave packet is prepared by a superposition of field-free
ground and excited state wave functions. The laser pulse is 1600 nm, 1014 W/cm2 , 6 fs FWHM. The
carrier-envelope phase is fixed (see text). Compared to when only the excited state is populated (dotted line
in (a)), the spectra have periodic intensity dips that show the attosecond wave packet motion. As the motion
becomes slower, the modulation period decreases.
1 × 10-16
1 × 10-18
1 × 10-20
1 × 10-22
1 × 10-16
1 × 10-18
1 × 10-20
1 × 10-22
1 × 10-16
1 × 10-18
1 × 10-20
1 × 10-22
of the previously split electron. Both contribute to the short-wavelength light that is emitted. They write
their joint signature on the light that the atom emits.
Figure 12 is a plot of the theoretical prediction of the light intensity emitted by a single atom. The
horizontal axis is the photon energy in electron volts. The vertical axis is a measure of the strength of
the light emission. In a real experiment, many atoms would contribute in a synchronized manner, and so
the light emission would be much more intense. The signal shown in Fig. 12 was calculated for the 6 fs
pulse shown on the right. It has a central wavelength of 1.6 µm, an intensity of 1014 W/cm2 . Figure 12
plots the results for wave packets with periods of 290, 330, and 444 as.
Prominent on each figure are periodic modulations of the light emission. It is possible to show that
the maxima appear when the two wave packets co-propagate. The minima appear when they counterpropagate. Therefore, a single image measures the wave packet motion [26].
© 2006 NRC Canada
Levesque and Corkum
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5. Conclusion
We have summarized some of the major trends in attosecond science and technology. It is clear
that attosecond technology is a radical departure from the ultrafast technology that preceded it. The
new technology science offers new opportunities for both optical science (attosecond pulses and highharmonic generation) and collision science (time-resolved measurements). Many aspects of the two
subjects can be fused, open completely new areas for investigation — such as orbital tomography.
The fused technology is a mixture of optical and collision science where the mutual coherence of the
electrons and photons can play a large role.
In essence attosecond science allows measurements at the most fundamental time scale for atoms
and molecules — the atomic unit of time —- combined with measurement at the most fundamental
space scale for atoms and molecules — the atomic unit of length.
Acknowledgements
Many people from NRC’s femtosecond science group (and elsewhere) have contributed key ideas
and experiments to the work that we have covered. While their specific contributions are mentioned in
the references, references alone can never show the intellectual team work that underlies such research.
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