Work supported in part by US Department of Energy contract DE-AC02-76SF00515
High
Harmonic Generation from Multiple Orbitals
in N
2
Brian K. McFarland, Joseph P. Farrell, Philip H. Bucksbaum, Markus G
¨
uhr
∗
PULSE Institute, Stanford Linear Accelerator Center, Menlo Park, CA 94025, USA
and Physics Department, Stanford University, Stanford, CA 94305, USA
∗
To whom correspondence should be addressed; E-mail: mguehr@stanford.edu
Molecular electronic states energetically below the highest occupied molecular orbital
(HOMO) should contribute to laser-driven high harmonic generation (HHG), but this
behavior has not been observed previously. Our measurements of the HHG spectrum
of N
2
molecules aligned perpendicular to the laser polarization show a maximum at the
rotational half revival. This feature indicates the influence of electrons occupying the
orbital just below the N
2
HOMO, referred to as the HOMO–1. Such observations of
lower-lying orbitals are essential to understanding the sub-femtosecond/sub-angstrom
electronic motion in laser excited molecules.
Tomographic imaging of molecules using high harmonic generation (HHG) has attracted wide inter-
est [1]. The method can be easily described in the framework of a strong-field three-step model [2, 3].
In this model, a portion of the electron wave function corresponding to the highest occupied molecular
orbital (HOMO) tunnels into the continuum and is accelerated in a strong oscillating optical field. This
continuum part of the wave function is treated as a free electron wave packet, which interferes coherently
with the bound part of the HOMO when it returns to the molecule. Recombination dipole radiation is
emitted on every half-cycle of the driving field and the coherent superposition of this radiation over mul-
tiple cycles forms a discrete spectrum of odd-order high harmonics. The spectrum contains information
about the HOMO structure. Tomographic reconstruction achieves sub-angstrom spatial resolution de-
1
SLAC-PUB-13406
September 2008
Published in the Science
termined
by the shortest de Broglie wavelength of the recombining electrons. The harmonics also carry
sub-femtosecond timing information due to the sub-cycle electron recombination. This high temporal
and spatial resolution can enable ultrafast movies of molecular orbital dynamics; however to date, only
the stationary HOMO has been imaged. Electron dynamics result from the coherent superposition of
multiple electronic stationary states. Observing sub-femtosecond electron dynamics therefore requires
the participation of multiple orbitals in HHG.
Here we report simultaneous HHG from two molecular electronic orbitals, the HOMO and the next
lower bound HOMO–1 in N
2
. The HOMO–1 participation emerges through enhancements in the HHG
signal at characteristic alignment angles of the molecular axis to the polarization of the harmonic gen-
erating pulse. The angular alignment is accomplished by impulsive rotational excitation of a cold (40K)
and dense N
2
jet by a nonresonant femtosecond laser pulse [4]. In the vicinity of a rotational revival,
the molecules undergo a rapid change in alignment with respect to the polarization of a second laser
which produces HHG. Although in our experiment low order harmonics (15-25) show a weakening of
the harmonic signal if the molecular axis is perpendicular to the harmonic generation polarization, at
higher harmonic orders a peak appears that is highly pronounced in this configuration. We attribute the
minimum amplitude in the low harmonics to HHG derived predominantly from the HOMO; the peak at
higher harmonics is characteristic of HHG from the HOMO–1. The prominent peak at higher harmonics
shows three important features of the HOMO–1 that are predicted by semiclassical simulations and sim-
ple ionization arguments: the cutoff extends to shorter wavelengths compared to an isotropic ensemble; it
is strongest when the molecular axes are near 90
o
to the harmonic generation polarization; and it presents
itself most clearly near the cutoff.
Past studies of HHG from aligned N
2
concentrated on the plateau region for orbital reconstruction
[1] and not on the cutoff region where the HOMO–1 signal is most pronounced. We observe that the
HOMO–1 signal is very sensitive to the molecular alignment and we tune our alignment parameters to
optimize the signal.
2
The
preferential field ionization of a more deeply bound state over a less deeply bound state is
possible because of the different geometries of the corresponding wave functions in the molecular frame.
This scenario is analogous to the field ionization of Stark states in atoms, where m
l
= 0 states ionize
more easily than m
l
> 0 states because the m
l
= 0 states overlap the saddle-point in the potential
[5]. Field ionization is most sensitive to the outer parts of the electron wave function near this saddle
point [6]. Figure 1A illustrates this point with cuts through isoprobability density surfaces of the HOMO
(solid line) and HOMO–1 (dashed line) of N
2
. The color indicates the relative sign of the wave function
amplitude. The orbitals are obtained by ab-initio calculations using an STO-3G basis set in the Gaussian
software package [7]. The N
2
HOMO and HOMO–1 exhibit σ
g
and π
u
symmetries, respectively. The
cuts through the HOMO and HOMO–1 orbitals show that the HOMO extends farther than the HOMO–1
along the direction parallel to the internuclear axis. Therefore the HOMO will preferentially ionize when
the electric field is parallel to the internuclear axis. The situation is reversed in the direction perpendicular
to the internuclear axis, where the HOMO–1 protrudes farther than the HOMO. This property is further
demonstrated in Fig. 1B which gives 1D cuts through the orbitals along the black dashed line in Fig.
1A. We conclude that the lower bound HOMO–1 ionizes more easily than the HOMO when the laser
field is polarized 90
o
with respect to the internuclear axis. This conclusion is supported by a strong field
ionization calculation for the HOMO and HOMO–1 orbitals [8].
Following strong-field ionization of a particular orbital, the acceleration of the electron in the laser
field results in an energy dispersed (chirped) electron wave packet. The recombining electron wave
packet is therefore a de Broglie wave, Φ
free
, whose instantaneous wavelength λ
dB
changes as a function
of recombination time. The returning electron wave forms a superposition with the N
2
orbital, inducing
a time dependent dipole moment described by hHOMO|ez|Φ
free
i or hHO M O–1|ez|Φ
free
i for the
respective orbital, as it moves across the N
2
molecule. Figure 1C sketches the recombination step for
molecules standing parallel to the harmonic generation polarization. Recombination to the HOMO gives
rise to a large dipole since the expectation value of z can vary all along the long axis of the HOMO [1].
3
The
HOMO–1, however, will not produce any signal because the nodal structure reduces recombination
probability [9] and, in addition, the opposite parity of the two HOMO–1 lobes produces two dipoles that
cancel because of a π phase shift.
For recombination with the generation laser polarization perpendicular to the internuclear axis (Fig.
1D), the electron distribution of the HOMO is considerably more confined and yields a diminished dipole
moment compared with the parallel case. The HOMO–1 on the other hand is less confined than the
HOMO and will give rise to a correspondingly stronger time dependent dipole moment. We neglect
dipoles perpendicular to the harmonic generation polarization because they are not phase matched for
alignment ensembles with mirror symmetry around the plane formed by the harmonic generation polar-
ization and propagation vector of the laser pulse [10].
In order to optimize the HOMO–1 signal, the molecular axes need to be aligned perpendicular to the
generation field. We can achieve this through impulsive alignment in two different ways: The alignment
and generation polarizations can be set parallel, which leads to an anti-aligned ensemble just past the
half revival (4.4 ps) where the molecules are perpendicular to the laser polarization. Alternatively, the
laser polarizations can be perpendicular in which case a prolate (see Fig. 2C) ensemble with the desired
alignment occurs at the half revival (4.1 ps). We found that an alignment scheme with perpendicular
polarizations enhances the contrast of the HOMO–1 with respect to the HOMO.
The HOMO–1 has a characteristic spectral signature. Due to the larger ionization potential (IP) of
the HOMO–1, we expect the HOMO–1 to have a higher cutoff than the HOMO. The HHG cutoff is given
by 3.17U
p
+IP, where U
p
is the ponderomotive energy [2].
Figure 2A shows a diagram of the experiment. We split the output of a 1kHz, 30 fs Ti:Sapphire laser
system into two time-delayed pulses using a wavefront beam splitter. The first pulse (alignment pulse,
90 fs for optimal alignment, intensity I
align
= 2.5×10
13
W/cm
2
) aligns the molecules, and the second
(generation pulse, 30 fs, intensity I
G
= 1.7 to 2.3×10
14
W/cm
2
) generates harmonics from the aligned
molecules. The pulses are focused with a spherical mirror (R = 800 mm) into a supersonically cooled
4
g
as jet of N
2
molecules in vacuum. The jet is positioned about 2 mm beyond the focus for optimal phase
matching [11]. The long alignment pulse excites a rotational wave packet in the molecules, leading to
periodic field-free alignment [4]. We examine the high harmonics generated near the half revival around
4.1 ps where constructive interference of the rotational coherence for N
2
leads to a strong modulation of
the alignment parameter hcos
4
θi [12]. Here θ is the angle between the alignment laser polarization and
the internuclear axis. The lower part of Fig. 2B shows the calculated value of hcos
4
θi for a rotational
temperature of T = 40 K, and an alignment laser intensity of I
align
= 2.5×10
13
W/cm
2
. The harmonics
between 20 and 70 eV pass through an Al filter (thickness 100 nm) into an imaging spectrometer where
the harmonics that are phase matched on axis (“short” trajectories) are selected by an aperture [13].
Figure 2B shows a false color plot of harmonic spectra collected for different delays between the
perpendicularly polarized alignment pulse and the harmonic generation pulse. The odd harmonics of the
fundamental radiation from the 15th to the 39th (cutoff) harmonic are modulated along the time axis by
the rotational revivals. The character of this modulation is similar for the lower harmonics but changes
for harmonics in the cutoff region. Specifically, the signal suppression in the plateau harmonics near
0.25 ps and 4.1 ps becomes an enhancement for cutoff harmonics.
We integrate over the center portion of each harmonic peak and normalize to the baseline alignment
signal. The resulting integrated harmonic signals are plotted from 3.5 to 4.7 ps time delays in Fig. 3A
for three values of the generation intensity I
G
. Concentrating on I
G
= 2.3×10
14
W/cm
2
, revival traces
are plotted for harmonic 15, and 25 to 39, with the cutoff at harmonic 39. Harmonics 15 to 23 have
the same shape, and we plot harmonic 15 as a representative of these curves. Starting at harmonic 25,
the minimum at 4.1 ps begins to flatten and harmonics 31 and greater show a peak superimposed on the
minimum. At harmonic 39, the temporal structure is completely inverted compared with the trend seen
for harmonic 15.
A comparison with the hcos
4
θi characteristics in Fig. 2B shows that the harmonic signal is mod-
ulated by the molecular revival structure induced by the alignment pulse. The 15th to 25th harmonics
5
