LETTER
doi:10.1038/nature14517
Linking high harmonics from gases and solids
G. Vampa
1
, T. J. Hammond
1
, N. Thire
´
2
, B. E. Schmidt
2
,F.Le
´
gare
´
2
, C. R. McDonald
1
, T. Brabec
1
& P. B. Corkum
1,3
When intense light interacts with an atomic gas, recollision
between an ionizing electron and its parent ion
1
creates high-order
harmonics of the fundamental laser frequency
2
. This sub-cycle
effect generates coherent soft X-rays
3
and attosecond pulses
4
,
and provides a means to image molecular orbitals
5
. Recently, high
harmonics have been generated from bulk crystals
6,7
, but what
mechanism
8–12
dominates the emission remains uncertain. To
resolve this issue, we adapt measurement methods from gas-phase
research
13,14
to solid zinc oxide driven by mid-infrared laser fields
of 0.25 volts per a
˚
ngstro
¨
m. We find that when we alter the genera-
tion process with a second-harmonic beam, the modified harmonic
spectrum bears the signature of a generalized recollision between
an electron and its associated hole
11
. In addition, we find that solid-
state high harmonics are perturbed by fields so weak that they are
present in conventional electronic circuits, thus opening a route to
integrate electronics with attosecond and high-harmonic techno-
logy. Future experiments will permit the band structure of a solid
15
to be tomographically reconstructed.
Fifty years ago, Keldysh introduced an important idea. Treating
both gases and solids on the same footing, he showed that strong-field
ionization—valence-band to conduction-band transitions in solids—
can be approximated by tunnelling
16
. Over the following decades,
research on the interaction of intense pulses with solids and gases
has diverged. Much of the solid-state research has been motivated
by laser materials processing
17
. A great deal of the gas-phase research
has been motivated by high-harmonic generation
18,19
, a process
initiated by tunnelling and completed by the recollision of a highly
energetic electron with its associated ion (hole). However, the deep
similarity in their response to strong fields has been largely ignored.
Now that high harmonics have been measured from ZnO (ref. 6) and
other condensed media
7,20
, it is important to reconsider the relation
between the gas-phase and solid-state mechanisms.
Individual ionizing atoms provide two sources of harmonics.
Expressed in the language of solid-state studies, there is a ‘single-band’
(intraband) contribution, in which the recently tunnelled electron
wave packet undergoes oscillatory motion in the laser field. Here the
nonlinearity enters through the step-wisenature of tunnel ionization
21
.
In addition, there is a ‘two-band’ (interband) source. In atoms, this
source is interpreted in terms of the ionized electron recolliding and
recombining with its associated hole (parent ion), emitting a high-
energy photon in the process. In gases, the former source dominates
the low harmonics while the latter source dominates high-harmonic
generation. Electrons and holes in solidsexhibit similar mechanisms as
they move in, and between, the conduction and valence bands
10,11
.
However, in a solid, the electron’s interaction with the lattice makes
its oscillation nonlinear; this is a new source of intraband emission
8
,
and leads to a generalized recollision
11
. The relative contribution of
these intra- and interband mechanisms and their interplay is import-
ant theoretically
8–12
and experimentally
7
. The unresolved issue is under
what circumstances does one physical mechanism dominate produc-
tion of high harmonics in solids.
To resolve this issue, we adapt a method of gas-phase research to
solids
13
. Using a mid-infrared fundamental laser to generate odd
harmonics in ZnO, we perturb the process with a very weak second-
harmonic beam, thereby producing even harmonics. The strength of
the even harmonics modulates as the second harmonic is delayed
relative to the fundamental. The phase of the modulation for each
harmonic order determines the spectral phase of the emitted harmonic
beam and characterizes the generation mechanism. Using photons
with energies well below that of resonance (hn < 0.1E
g
, where E
g
is
the bandgap of the solid), we find that a generalized recollision
between the electron and its associated hole plays the dominant role
in high-harmonic emission from ZnO at the field strength of our
experiment, as it does in gases. By ‘generalized recollision’ we mean
that between tunnelling and recollision the electron and hole move
on their respective bands, as discussed in the Methods section.
Furthermore, as we increase the relative second-harmonic intensity,
for each intensity range we find solid-state behaviour that closely
mirrors the atomic response. There is, however, one possible excep-
tion. As described in the Methods section, in gases there are so-called
long and shorttrajectory contributions, which lead to differently diver-
ging radiation. Although we have searched for them, we have not
resolved long-trajectory harmonics in our experiment. However, even
this may be the same in gases since long-trajectory harmonics are hard
to observe with infrared drivers.
To understand the experiment it is useful to think of high-harmonic
generation as a balanced interferometer. As illustrated in Fig. 1a for
atomic recollision, the electron (and hole) wave packets created on
subsequent half-cycles of the driving field propagate on equal but
oppositely directed trajectories (equal-length arms of an interfero-
meter), creating sub-cycle bursts of short-wavelength radiation when
they recollide. With each subsequent pulse having opposite phase
because of the oppositely directed collision, when the pulses are spec-
trally resolved, only odd harmonics constructively interfere. Adding a
weak second-harmonic field unbalances the trajectories (interfero-
meter), adding a small phase to one arm while removing it from the
other, thereby producing even harmonics. For ZnO, because of the low
reduced carrier mass near the band centre, low-photon-energy har-
monics are very sensitive to weak second-harmonic fields. In a recolli-
sion process, the phase that maximally unbalances the interferometer
differs between harmonic orders. While illustrated for recollision, the
interferometer analogy also applies to intraband oscillations: here,
the asymmetry originates from the slightly different current generated
in two successive half-cycles, as a result of the electron reaching
higher (lower) momentum when it propagates in the left (right) side
of the Brillouin zone. Because harmonics are emitted as the electron
moves, they are all phased together
10
. The second harmonic perturbs
all of them equally and simultaneously, so the phase that maximally
unbalances the interferometer is the same for all harmonics. This
different behaviour allows us to discriminate between recollision and
non-recollision based mechanisms.
Experimentally, high harmonics are produced by focusing the
fundamental and the second harmonic of mid-infrared laser pulses
(3.76 mm central wavelength) in a 500-nm ZnO single crystal at an
intensityof 0.85 TW cm
22
(0.25 V Å
´
21
) in the crystal. The two colours
462 | NATURE | VOL 522 | 25 JUNE 2015
1
Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada.
2
INRS-EMT, 1650 boulevard Lionel-Boulet, CP 1020, Varennes, Que
´
bec J3X 1S2, Canada.
3
National Research Council of
Canada, Ottawa, Ontario K1A 0R6, Canada.
G
2015 Macmillan Publishers Limited. All rights reserved
(that is, fundamentaland second harmonic) have parallel polarizations
(details of the experimental set-up are given in the Methods section).
Figure 1b shows the harmonic spectrum versus the relative phase
between the two colours, for a second-harmonic intensity 9 3 10
26
times that of the fundamental. Each harmonic order is independently
normalized; the relative intensity between adjacent orders (I
2N
/I
2N21
)
is reported at the top of the graph (N is an integer that identifies the
harmonic order). The weak even-harmonic signal modulates with an
order-dependent phase W
osc
(2N) as a function of delay. Similar beha-
viour is measured for atomicharmonics
13
, where W
osc
is used to extract
the spectral phase of each even harmonic or, equivalently, the emission
time of the attosecond pulses. Here, we use W
osc
to establish the origin
of solid-state harmonics.
Figure 2 shows W
osc
plotted as a function of harmonic order (black
circles) up to the cut-off of our spectrograph. We use an unmeasured
constant phase to position the experimental data (this constant phase is
measurable using nonlinear optical methods). The theoretical values of
W
osc
calculated from a two-band model are also plotted for the intra-
band (purple line) and interband (blue line) sources, and for their
combined emission (gold dashed line); the calculations are described
in the Methods section. We find that the experiment is consistent with
the theoretical prediction (Extended Data Fig. 6) that interband har-
monics dominate in ZnOfor the field strengthused (we notethat recent
work has found evidence for the dominance of intraband oscillations
forTHz excitation
7
). In the Methodssection weinvestigate theoretically
how our results scale with the field strength of the fundamental.
The slope of the curves in Fig. 2 is determined by both the intensity
of the fundamental and the band structure of the crystal. Crystal dis-
persion has two major consequences:first, the electron–hole pair accu-
mulatesmore phase than in a gas (,10times for the short trajectories),
making it easier to break the symmetry with the second-harmonic
field. This is why we require second-harmonic intensities of only
,10
25
times that of the fundamental. Second, crystal dispersion influ-
ences the duration of the high-harmonic pulses. We measure a har-
monic chirp of 0.38 fs eV
21
at the 16th harmonic, corresponding to a
train of pulses with durationof 1.7 fs (before dispersion compensation)
for a ,2 eV bandwidth. Wider bands and higher intensities will
result in smaller chirps, thereby allowing attosecond pulse generation
from solids.
As the second-harmonic intensity is increased, we move from per-
turbing to controlling harmonic emission. In gases, we first reach an
intensity where W
osc
is nearly out of phase for the even and odd
harmonics
13
. Figure 3a shows experimental results for ZnO using a
second-harmonic intensity of ,10
24
times that of the fundamental.
The alternation between even and odd harmonics occurs when the
second harmonic modulates the electron phase by W
osc
5 p/2. For
even higher second-harmonic field strengths (Fig. 3b), high-harmonic
emission is substantially modified. In the highly asymmetric sum field,
tunnelling only occurs once per laser cycle. This leads to simultaneous
emission (or suppression) of even and odd harmonics—a behaviour
well-studied for atomic gases
14
.
Taken together, our findings provide strong evidence that, for the
field strength and laser wavelength of our experiment, a generalized
recollision between an electron and its associated hole is the primary
source of high harmonics in ZnO. Thus, 50 years after Keldysh’s
seminal paper
16
, we rebuild the connection linking the high-field res-
ponse of gases and solids. But decoherence is a major difference
between solids and gases. The occurrence of electron–hole recombina-
tion demonstrates the tolerance of the short-lived electron to dephas-
ing. Although scattering times vary considerably between materials
22,23
and with excitation conditions
24
, our measurements suggest that high-
harmonic technology could have a broad impact in condensed media.
In this paper we have exploited our ability to manipulate the phase
that the electron–hole pair acquires to identify the fundamental mech-
anism dominating harmonic generation in ZnO. The phase also
encodes information about the band structure of the material in which
the electron travelled. By measuring W
osc
asafunctionofcrystalori-
entation, it willbe possible to reconstruct a material’s three-dimensional
momentum-dependent band structure
15
. This all-optical technique will
allow measurement of band structures where it is not possible to detect
photoelectrons, for example in high-pressure experiments
25
.
Although we have not concentrated on it here, the tunnelling step
determines the harmonic intensity. Our weak second-harmonic beam
61014182226
3
4
5
6
7
Harmonic order
osc
(2N) (rad)
Bandgap
Experiment
ZnO, interband
ZnO, intraband
ZnO, total
Φ
Figure 2
|
Comparison with theory. The phase of the modulation of the
even harmonics is extracted from Fig. 1 (black circles) and compared to
the simulated intraband (purple line) and interband (blue line) phase. The
simulated phase for their combined emission (intraband plus interband,
yellow dashed line) agrees with the interband emission, in agreement with
interband harmonics being significantly stronger than intraband ones
(Extended Data Fig. 6).
6 8 10 12 14 16 18
0
0.5
1
1.5
2
Harmonic order
Delay (cycles)
a
b
0.12 0.04 0.12 0.07 0.1 0.11 0.14
I
2N
/I
2N–1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Spectral intensity
Figure 1
|
Realization of the
in situ
experiment. a, Pictorial representation of
the electron’s trajectories. In the fundamental field alone, the electron travels
symmetrically to the left and to the right of the ion (central grey sphere) during
two successive half-cycles (red dashed lines). The addition of the second
harmonic lengthens one trajectory while shortening the other (solid blue lines),
thereby breaking the symmetry and producing even harmonics. b, Measured
high-harmonic spectra versus the delay between the two colours; the spectral
intensity is colour-coded (key at right). The even harmonic intensity modulates
every half-cycle. The odd harmonics also modulate, but the modulation is
much weaker. The phase of the even harmonic modulation depends on the
harmonic order, as evidenced by the red solid line relative to the black reference
line. (The red line links the minima in the modulation of the even harmonic
intensity.) Each harmonic order is normalized separately. The relative
intensity between even-order and adjacent odd-order harmonics is given at the
top (I
2N
/I
2N21
). The delay of the second harmonic is defined in cycles of the
fundamental.
25 JUNE 2015 | VOL 522 | NATURE | 463
LETTER RESEARCH
G
2015 Macmillan Publishers Limited. All rights reserved
not only modifies the electron phase but also slightly alters tunnelling,
thereby leading to oscillations of the odd harmonics with the delay
between the two colours. The relativephase of the modulation between
even and odd orders contains detailed information on the time-
dependent tunnelling rate
26
. Applying this analysis to solids will allow
us to probe the sub-cycle dynamics of strong-field tunnelling in
solids
27,28
—the essence of laser material modification
17
.
Finally, independentof the mechanism, we havedemonstrated thata
clear spectralsignature canbe imposed on a high-harmonic beamusing
a control beam having a peak electric field of ,5Vmm
21
.Fieldsofthis
magnitude are present in electronic circuits
29
. Thus, we have bridged
the gap between electronics and attosecond physics. If the sensitivity
that we have demonstrated to perturbing fields can be transferred to
silicon, it will be possible to record movies of working semiconductor
electrical circuits or plasmons propagating in nano-plasmonic devices.
Harmonics generated from a spatially complex circuit will diffract
according to the instantaneous distribution of the internal fields. The
control field can also be applied by electrodes on the crystal, thereby
allowing manipulation of the harmonic beam with DC or pulsed elec-
trical signals. Thus we have taken a significant step towards a new area
of research, that of solid-state attosecond electronics
30
.
Online Content Methods, along with any additional Extended Data display items
and Source Data,are available in the online version of the paper; references unique
to these sections appear only in the online paper.
Received 11 November 2014; accepted 27 April 2015.
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Acknowledgements We thank A. Laramee
´
from the Advanced Laser Light Source for
technical support during the experiment and M. Clerici for lending some equipment.
We acknowledge financial support from the US AFOSR,NSERC, FRQNT, MDEIE and CFI.
Author Contributions G.V. and P.B.C. conceived the experiment; G.V., T.J.H. and N.T.
performed the experiment; N.T. and B.E.S. developed the laser source; P.B.C. and F.L.
supervised the experiment; C.R.M. and T.B. supervised the theoretical calculations; all
authors contributed to the manuscript.
Author Information Reprints and permissions information is available at
www.nature.com/reprints. The authors declare no competing financial interests.
Readers are welcome to comment on the online version of the paper. Correspondence
and requests for materials should be addressed to P.B.C. (paul.corkum@nrc.ca) or G.V.
(gvamp015@uottawa.ca).
Harmonic order
0.69 0.58 0.37 0.41 1.5 0.3 1.1 0.68 0.67 0.53 0.490.35 0.20.2
I
2N
/I
2N–1
a
6 8 10 12 14 16 18
0
0.5
1.0
1.5
2.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Harmonic order
b
6 8 10 12 14 16 18
0
0.5
1.0
1.5
2.0
Delay (cycles)
Spectral intensity
Figure 3
|
From perturbing to controlling harmonic generation. a, When
the intensity of the second harmonic reaches 1 3 10
24
times that of the
fundamental, the even and odd harmonics modulate out of phase. b, Further
increasing the intensity to 3 3 10
23
times that of the fundamental, we find that
all harmonics modulate in phase. Each harmonic order is normalized
separately. The relative intensity between even-order and adjacent odd-order
harmonics is given at the top.
464|NATURE|VOL522|25JUNE2015
RESEARCH LETTER
G
2015 Macmillan Publishers Limited. All rights reserved
METHODS
Experimental set-up. High harmonics are generated with a mid-infrared laser
source centred around 3.76 mm and with 19 mJ pulse energy. The pulse duration of
95 fs is measured with a dispersion-free SHG FROG after the ZnO crystal.
The mid-infrared wavelength is obtained by difference-frequency generation in
a 400 mm thick AgGaS
2
(AGS) crystal between the signal and the idler of a
commercial Optical Parametric Amplifier (Light Conversion Topas-HE) pumped
by 4.5 mJ from a Ti-sapphire laser. The output of the OPA is further amplified in a
beta barium borate (BBO) crystal pumped by 10 mJ of 800 nm pulses. The mid-
infrared beam is spatially filtered with a pin-hole and focused with an f/30 spher-
ical Ag mirror onto an epitaxially grown 500 nm thin film of a single crystal of
wurtzite ZnO(0001) deposited on a 0.5 mm sapphire(0001) substrate. The optical
axes of both crystals are aligned parallel to the laser
k
-vector. We used the knife-
edge technique to measure the size of the focal spot.
The second harmonic is generated in a 300 mm AGS crystal optimized for type-I
SHG right after the pin-hole to exploit the high intensity and high beam quality.
The second-harmonic intensity is controlled by moving the AGS crystal closer to
or farther from the pin-hole. The two beams are separated and then recombined
before the focusing mirror with dichroic beam splitters (LaserOptick) that reflect
the second harmonic. Its polarization is rotated with a broadband l/2 plate before
recombination. The delay between the two colours is scanned with a PZT stage on
the second-harmonic arm.
The high harmonics are refocused with an Al mirror into a visible–ultraviolet
spectrometer from OceanOptics.
The theoretical model. To interpret the experiment, we use the theoretical frame-
work developed in ref. 11. The nonlinear response of the solid is described by a two
band model whose characteristics are chosen to be those of ZnO. The model solves
the semiconductor Bloch equations for the time-dependent band populations
n
m
(
k
,t)(m 5 v, c for the valence and conduction bands respectively) and for the
interband polarization p(
k
, t), where
k
is the crystal momentum, for a single
electron–hole pair. Coulomb interaction between different pairs, leading to coup-
ling of different k-states
31
, is neglected. The band parameters and lattice constants
are reported in ref. 32. The laser parameters are: frequency v 5 0.0121 a.u., peak
field strength F
0
5 0.0049 a.u.
Extended Data Fig. 1 shows the population of the conduction band along the
CM direction of the reciprocal space as a function of time. The laser field creates
electron–hole pairs around each field extreme. They are then accelerated to high
crystal momenta and back to the centre of the Brillouin zone in a process that
repeats every half laser cycle.
The model naturally includes the sub-cycle exchange of the band population—
the analogue of the step-wise ionization in gases. However, in solids both the
single-band (intraband) and the two-band (interband) mechanism bear the sig-
nature of the generalized electron–hole recollision.
Extended Data Fig. 2a, b shows the high-harmonic spectrum of the respective
intraband and interband terms as a function of time. The spectra are obtained
from a windowed Fourier transform of the intra- and interband currents, defined
in ref. 11, which can be separately calculated by the model. A 0.35 cycles wide
Blackman window is scanned across half a laser cycle. The resulting spectra are
continuous because only one recollision event is allowed inside the temporal
window. We apply a super-Gaussian spectral filter that does not affect above
bandgap harmonics, but progressively filters lower harmonics. This is done
because the temporal window is too narrow for the lower orders (and the fun-
damental frequency) and results in artificial broadening of these harmonics which
ultimately masks the weak high-harmonic signal if the filter is not applied.
Interband harmonics above the bandgap are approximately 10
4
times stronger
than intraband ones. For each harmonic photon energy, the maximum spectral
intensity determines its time-of-emission (or spectral phase).
For comparison, the time of generalized recollision is traced in red in Extended
Data Fig. 2a, b. The time of re-encounter of the electron with its associated hole is
extracted from their classical motion in the conduction (for the electron) and
valence (for the hole) bands relative to the time of the field crest in which they
were created at zero crystal momentum by strong-field tunnelling. Their accelera-
tion in reciprocal space is
k
(t) 5
A
(t) 2
A
(t9), where
A
(t) is the laser vector
potential, and t9 is the time of creation of the electron–hole pair. The real space
velocity is
v
m
[
k
] 5=
k
E
m
(
k
), with E
m
(
k
) the energy dispersion for band m, from
which the instantaneous position is obtained by temporal integration. This semi-
classical model, introduced in ref. 11, has been detailed in ref. 32.
Extended Data Fig. 2 shows thatbothinterband and intraband currents produce
high harmonics with a spectral phase determined by the generalized recollision.
This recollision contribution is contained in the structure of the intraband equa-
tion
11
.
Contrary to interband harmonics, intraband emission is dominated by
nonlinear oscillations and by tunnelling below the 18th harmonic order. The
transition is reflected in the in situ calculation (see Fig. 2), where W
osc
becomes
independent on the delay between fundamental and second harmonic below the
18th harmonic.
Many electron–hole pairs are created by the strong laser field. Owing to the
interaction of one electron (hole) with the others, the electron (hole) can scatter to
a different k-state than that of its correlated hole (electron), therefore preventing
their recombination. However, in a many-body theory, each carrier can encounter
a different oppositely charged partner. In atomic high harmonic generation it was
argued that recombination of the electron with an uncorrelated hole is not allowed
in principle
33
. This debate is yet to be settled in solids. Our experiment only detects
recollision between correlated electron–hole pairs.
Modelling the experiment. A weak second-harmonic field is added to the fun-
damental one to simulate the in situ experiment. The high-harmonic spectrum is
recorded for each phase between the two colours. The resulting spectrograms for
the interband and intraband emission are reported in Extended Data Fig. 3a, b,
respectively. We require the second harmonic to be 6 3 10
24
relative to the
fundamental to produce even high harmonics that are ,5% of the odd ones.
The modulation depth is close to 100% for both intra- and interband generated
above bandgap harmonics. It drops to ,50% for below bandgap interband har-
monics and for harmonics below the 18th for intraband emission. The spectro-
grams of the interband and intraband currents and of the sum of the two are
analysed in the same way as the experimental data: (i) the intensity of each
harmonic is spectrally integrated over its width; (ii) the resulting intensity modu-
lation as a function of delay is offset to 0 and normalized to 1 and (iii) fitted to a
cosine function of delay with form cos(w
delay
1 W
osc
). Finally, W
osc
for both
emission mechanisms and for the total emission (the sum of the two) is extracted
and shown in Fig. 2 of the main text.
Comparison with atomic high-harmonic generation. We analyse the similar-
ities between the high harmonics from atoms and solids based on calculation of
classical trajectories. In Extended Data Fig. 4 we plot W
osc
calculated for an isolated
atom (green line). The field strength of the fundamental is increased to 0.44 V Å
´
21
to allow the same cut-off as in ZnO. The phase for interband emission in ZnO and
the experimental data are the blue line and the black circles, respectively.
According to the analytical equivalence between atomic and solid emission,
demonstrated in refs 11, 32, the intensity modulation of the even harmonics can
be calculated starting from the additional dipole phase
13
:
s(t,w)~
ð
t
t’(t)
v(t,t’)A
2
(t,w)dt
Where
A
2
(t, w) 5
A
2
cos(2vt 1 w) is the vector potential of the second harmonic
and
v
(
k
) 5=
k
e
g
(
k
), with e
g
(
k
) the momentum dependent bandgap. Once the
semi-classical trajectories are known (as discussed in the previous section),
the integral can be computed and the phase of the modulation extracted from
13
W
osc
5 atan(s
c
/s
s
) with s(t, w) 5 s
s
(t) cos(w) 1 s
c
(t) sin(w).
In atomic high-harmonic generation there are two important contributions to a
given high harmonic. One is from the so-called short trajectory electrons. They
appear as the curve with negative slope at larger phases in Extended Data Fig. 4.
Almost all research on gas-phase harmonics has concentrated on this contri-
bution. There are also long trajectory electrons. They are responsible for the other
part of the curve and are often ignored in gases since their divergence is larger and
an experiment can be designed to minimize their impact.
In ZnO, the long trajectory contribution is significantly different from the
atomic case. This is a direct consequence of the non-parabolic band dispersion.
The distortion of the trajectories is extensively analysed in ref. 32. Further, no
prediction can be made for below-bandgap harmonics, since classically the elec-
tron cannot recollidewith energy smaller than the minimum bandgap.A quantum
mechanical calculation that includes the effect of the tunnelling step is required to
extend the comparison to smaller photon energies.
Theoretical cut-off scaling of interband emission. The high-harmonic spectrum
extendsup to a defined maximum harmonicorder, called the cut-off. In atoms, the
cut-off is related to the maximum kinetic energy available to the electron when it
recombines with the hole. The cut-off depends linearly on the ponderomotive
energy U
p
, which, in turn, depends quadratically on the field strength.
In contrast with atomic harmonics, experiments in solids
6
show a linear scaling
of the cut-offwith field strength. Theoretical investigation of the intraband current
confirmed the linear scaling
8
. Extended Data Fig. 5 shows that the simulated
interband current also leads to a cut-off that scales linearly with the field strength.
Exploiting the analytical solution for this current
32
, it is possible to derive a semi-
classical approximate cut-off scaling law that fully agrees with the simulation. This
analysis is reported in ref. 32.
Role of interband emission for various field strengths. We theoretically invest-
igate the relative importance of interband and intraband emission at different field
strengths at the laser wavelength used in this experiment. Extended Data Fig. 6
reports the interband and intraband spectra for F
0
5 0.003 a.u., F
0
5 0.0049 a.u.
LETTER RESEARCH
G
2015 Macmillan Publishers Limited. All rights reserved
(corresponding to the field strength reached in our experiment) and F
0
5 0.008 a.u.
(corresponding to the field strength reached in ref. 6, including the loss from
normal incidence reflection). In all cases, interband emission dominates by four
orders of magnitude for harmonic orders above the minimum bandgap (marked by
the vertical dashed black line). Below the minimum bandgap the difference is less.
A less clear harmonicstructure is observed at the highest field strength. The loss
of contrast arises from the interference between two sets of recolliding trajectories:
one of electrons born slightly after the peak of the field (the ‘short’ branch of the
atomic case) with one of electrons born just before the peak of the field. In
the atomic case, electrons born before the peak do not recollide. In a solid, the
non-parabolic shape of the band structure allows these extra recombinations. The
topic will be a matter of a future publication.
31. Golde, D., Kira, M., Meier, T. & Koch, S. W. Microscopic theory of the extremely
nonlinear terahertz response of semiconductors. Phys. Status Solidi B 248,
863–866 (2011).
32. Vampa, G., McDonald, C. R., Orlando, G., Corkum, P. B. & Brabec, T. Semiclassical
analysis of high harmonic generation in bulk crystals. Phys. Rev. B 91, 064302
(2015).
33. Niikura, H. et al. Probing molecular dynamics with attosecond resolution using
correlated wave packet pairs. Nature 421, 826–829 (2003).
RESEARCH LETTER
G
2015 Macmillan Publishers Limited. All rights reserved
