4872
OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
Spectral modulation of high-order harmonic generation
from prealigned CO2 molecules
Peng Peng, Na Li, Jiawei Li, Hua Yang, Peng Liu,* Ruxin Li, and Zhizhan Xu
State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics,
Chinese Academy of Sciences, Shanghai 201800, China
*Corresponding author: [email protected]
Received August 27, 2013; revised October 14, 2013; accepted October 17, 2013;
posted October 21, 2013 (Doc. ID 196396); published November 14, 2013
We demonstrate experimentally that prealigned molecules produce observable spectral redshift or blueshift on the
high-order harmonic generation. We distinguish two effects of molecular alignment on the phase modulation of the
harmonics; one is from the gradient of alignment degree and the other is the plasma density varied by the molecular
alignment. The finding provides an insight on the spectral distribution of molecular harmonics and a method of
fine-tuning the harmonic spectrum. © 2013 Optical Society of America
OCIS codes: (020.2649) Strong field laser physics; (350.5400) Plasmas.
http://dx.doi.org/10.1364/OL.38.004872
High-order harmonic generation (HHG) via intense femtosecond (fs) laser pulses provides a distinct source of
coherent extreme ultraviolet (XUV) radiation whose
duration can reach up to tens of attoseconds [1–3]. The
generation process of the high harmonics is composed of
the ultrashort dynamics of acceleration and recombination of the tunneling ionized electrons within an optical
cycle [4]. Therefore, HHG from field-free-aligned molecules has been applied to probe the structure of molecular orbitals and the nuclear dynamics of molecules in
subfemtosecond resolution [5–12]. Among the studies,
either the spatial distribution of the HHG emission from
aligned molecules or the temporal modulation of the
spectra after the aligning laser pulses were measured
to reveal the underlying physics of molecules in intense
laser fields.
The frequency of the HHG spectrum can be shifted by
the spectral phase modulation (SPM) which comes from
the modification of the refractive index of the medium,
owing to the optical Kerr effect and free electrons generated by the intense laser field [13–17]. It has been
shown that the aligned molecules could alter the spectral
distribution of the ultrafast fs laser pulses [18]. However,
the molecular alignment effect on the HHG spectrum has
not been elucidated so far. In the present work, we find
that the spectral distribution of high harmonic emission
is tuned by a pre-excited rotational wave packet of molecules. Both the blueshift and redshift of spectra can be
achieved through the selection of the evolution time of
the wave packet.
An excited rotational wave packet of molecules is
generated through the nonresonant impulsive Raman
process during the interaction between a short intense
laser pulse and the molecular anisotropic polarizability.
After the extinction of the field, the periodic rephasing
and dephasing of the rotational eigenstates leads to a
temporal revival structure of molecular alignment and
antialignment [19]. The emission intensity of HHG from
aligned molecules has been shown to modulate owing to
the two-center interference effects and/or multiorbital
contributions [7–9,20,21], but it is not known so far
how the molecular alignment impacts on the spectrum
of the HHG. Our investigation indicates that the temporal
0146-9592/13/224872-04$15.00/0
modulation of molecular alignment can manipulate
the spectral content of high-order harmonic emission
through two SPM mechanisms. One is the modulation
of the refractive index directly from the molecular alignment and the other is the variation of plasma density.
The experiments are performed using a Ti:sapphire
chirped pulse amplifier laser system (Coherent EliteHP-USX, 5 mJ∕25 fs∕1 kHz at the center wavelength of
800 nm). The output laser pulse is split into two beams:
one beam (40% in energy) is used as the pump pulse for
aligning the CO2 molecules and the other is in the same
polarization as the probe pulse for HHG. The two beams
are collinearly focused by a lens (f 300 mm) onto a
pulsed supersonic molecular beam located in a high vacuum chamber. The stagnation pressure of CO2 gas
(99.998% in a mole) is about 2 bars, and the laser focus
is located 2.0 mm in front of the molecular beam to ensure only the short trajectory of HHG is phase matched.
The laser pulses cross with the molecules at a distance of
0.8 mm from the nozzle orifice whose diameter is 0.5 mm.
The rotational temperature is estimated to be ∼60 Kelvin
(K) based on the molecular beam parameters [22]. The
probe pulse energy is about 0.9 mJ and the peak intensity
is estimated as ∼1.8 × 1014 Wcm−2 based on the highest
harmonic order measured by the probe pulses [4]. In order to optimize the alignment condition of the molecules,
the pump laser energy was adjusted to achieve the highest harmonic emission at the revivals using a half-wave
plate and a high extinction film polarizer. Under the optimized alignment condition, the pump laser field intensity
is estimated to be ∼6.0 × 1013 Wcm−2 . Each HHG spectrum is recorded by integrating the data from over 600
laser shots using a homemade flat-field grating spectrometer equipped with a soft x-ray CCD camera (Princeton
Instruments, PI: SX 400).
When the CO2 molecules are irradiated by the pump
pulses whose duration τon 25 fs is much shorter than
the rotational period of T rot 42.7 ps, nonadiabatic
field-free alignment is achieved by the excitation of a rotational wave packet ψt ΣJ;M AJ;M tjJ; Mi [19]. The
time evolution of the wave packet can be calculated by
solving the time-dependent Schrödinger equation (TDSE)
and the alignment parameter hcos2 θit is defined as
© 2013 Optical Society of America
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
hcos2 θit X
J
ρJ
X
hψtjcos2 θjψtiJ;M ;
4873
(1)
J;M
in which ρJ represents the initial Boltzmann distribution
of the molecules over all rotational states jJ; Mi. The
modulated spectra of high harmonics are recorded at
the continuous varied time delay after the probe pulses,
as shown in Fig. 1.
We focus on the modulation of molecular HHG of the
23rd–29th orders (H23–H29) at the quarter revival during
the delay time of 9 ps– ∼ 13 ps, where two characteristic
delays with similar amplitudes, but reversed derivatives
of alignment degree, can be easily selected. One can see
that the harmonic emission of H23–H29 represents a temporal modulation that is correlated to the molecular
alignment degree. Figure 1 also plots the integrated emissions of H23–H29 at each time delay of probe pulses and
the calculated molecular alignment degree versus the delay time. From the calculated alignment curve, the
residual alignment between the revivals is distinguishable as the alignment parameter reaches near 0.4 instead
of 0.33 for an isotropic distribution [23,24]. The modulation of harmonic intensity is seen reversely matching
with the alignment degree. The observation is consistent
with the previous studies and has been explained by
the two-center interference effect of the recombining
electron waves generated from the two oxygen atoms
in CO2 , therefore the inverted harmonic orders can be
a probe of the nuclear distance for molecules within
intense laser fields [7–9,12].
It is interesting to note that in Fig. 1 the spectra of the
harmonics are also varying at the time delays of the
evolving molecular alignment. Comparing the harmonic
spectrum at the time of 10.52 ps when the alignment degree is the maximum (alignment condition, delay A in
Fig. 1) with that at 11.04 ps when the alignment degree
is the minimum (antialignment condition, delay C), one
can see that the spectra of H23–H29 for antialigned molecules all extend to the red side much more than that for
the aligned molecules. For more detailed investigation,
the spectra of H27 at five delay times, labeled as A, B,
C, D, and E in Fig. 1, are retrieved and plotted in Fig. 2.
Fig. 2. Spectra of H27 at the delay times of A, B, C, D, and E
labeled in Fig. 1.
The five delay times are purposely selected to be characterized as the maximum alignment, the falling slope of
alignment degree, the minimum alignment, the rising
slope of alignment, and the relative isotropic alignment
of molecules, respectively. Evidently, the molecular
alignment condition can modify not only the intensity
of the harmonics, but also their spectral distribution.
For revealing the molecular alignment effect on the spectral distribution, the spectrum of H27 at the delay time E
is chosen as a reference because it is away from the
revival and the molecules are nearly isotropic at this moment. Since the pump pulse may induce weak ionization
of the molecules, the effect of the produced free electrons by the pump pulse can be excluded by using the
spectrum at E as a reference. Comparison of the harmonic spectra at the delay times of A, B, C, and D with
the reference spectrum can distinguish the effect of
molecular alignment from the others such as the nonadiabatic response and the propagation of the fundamental
pulses. As can be seen, the alignment condition at the
maximum degree (A) and the falling slope (B) induces
blueshift on the harmonic spectrum, while the minimum
alignment (C) and the rising slope (D) produce redshift,
compared with the spectrum at the delay time of E. The
observation is confirmed by the normalized spectra of
H27 at all the delay times.
The spectrum of harmonic emission is coherently
determined by temporal phase modulation, which originates from the variation of refractive index as the laser
pulses and the harmonics propagate through the nonlinear medium. Owing to the molecular alignment, the
refractive index is modified by [18]
ρ0 Δα
hcos2 θit − 1∕3;
2π
n0
δn∕∕
Fig. 1. Right side: harmonic spectra of H23–H31 around the
quarter revival of aligned CO2 molecules; Left side: the integrated harmonic intensity (red circular markers) and the calculated alignment parameter hcos2 θit (blue line) versus the
pump–probe delay.
(2)
in which ρ0 denotes the number density of molecules,
n0 is the linear refractive index and Δα is the difference between polarizability components along and
perpendicular to the molecular axis. As a result, the
spectral shift is given by the gradient of the phase variation of the propagating pulses,
4874
OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013
δλ λL dδn∕∕ λLρ0 Δα dhcos2 θit
2π
;
dt
c
cn0
dt
(3)
in which λ is the wavelength of the propagating pulse and
L is the interaction length. This indicates that the derivative of the alignment parameter results in the spectral
shift of the interacting pulses. The previous studies have
elucidated this effect on the propagating ultrafast laser
pulses [18]. For the HHG from molecules, the spectral
shift of H q (q is the harmonic order) is therefore
calculated by
Δλcal
q ≈
δλ1
λ Lρ Δα dhcos2 θit
;
δλq 4π 1 0
qcn0
dt
q
(4)
in which Δλcal
q is the calculated wavelength shift of H q , λ1
is the wavelength of the fundamental pulse, λq is the
wavelength of the H q . The first term on the right side
of Eq. (4) is the effect of spectral modulation of the fundamental pulse that is directly imprinted on the harmonic
spectrum, and the second term denotes the effect on the
harmonics propagating through the prealigned molecules. In our experiments, the interaction length and
number density are 1 mm and 1 × 1018 cm−3 , respectively.
For CO2 , n0 1.0005, and Δα 2.03 Å3 [25]. Therefore,
at the selected delay times of B and D, the spectral shifts
−3
nm, and
of H27 are calculated as Δλcal
27 B −3.0 × 10
cal
−3
Δλ27 D 2.6 × 10 nm; the minus sign indicates the
spectral blueshift. From the experimental result, one
can determine theRspectral position
using the averaged
R
wavelength, hλi λ · I λ dλ∕ I λ dλ, and the result shows
exp
that Δλexp
27 B −0.045 nm, and Δλ27 D 0.017 nm.
The calculation result indicates the redshift and blueshift
as shown in the experiment, but with an error of one
order of magnitude in the quantities. This discrepancy
may come from the effect of plasma that can be calculated based on the alignment degree and the estimated
laser intensity.
At the delay times of A and C, when the alignment degree is the maximum and minimum, the spectral shift effect of molecular alignment is suppressed because the
derivative of the alignment degree is zero. The observed
even larger spectral shift, Δλexp
27 A −0.058 nm, and
C
0.057
nm,
can
be
completely
attributed to
Δλexp
27
the changed number density of free electrons. The variation of refractive index is calculated by [14]
δn0 −
e2
N e;
2ε0 mω2
are in the varying alignment conditions. Considering that
the molecular axis is at an angle of θ with the laser polarization, the angular dependent ionization rate of CO2 ,
W θ, is calculated using the molecular Ammosov–
Delone–Krainov (MO ADK) theory under the laser field
of 1.8 × 1014 Wcm−2 . W t is then calculated by integrating the product of W t and the angular distribution of the
molecules, Pθ; t,R at an alignment condition modulated
by time, W t W θPθ; tdθ. As a result, the W t
for A and C are 0.20 and 0.03, respectively, compared
to the 0.14 for the time delay of E. The spectral shift
of H q is therefore calculated by combining the effects
of fundamental pulses and harmonic pulses
Δλcal
q
3 2
δλ01
λ1 e L ρ0 W t
1 1
0
δλq − 3
≈
:
q
q q 8π 2 ε0 mc3 τon
(7)
At the delay times of A and C, the spectral shifts
of H27 are calculated as Δλcal
27 A −0.071 nm, and
Δλcal
27 C 0.122 nm, which has the same order of magnitude as the experimental result. It is noted that the values are also close to the previous result obtained by
Wahlström et al. [14] on xenon in similar conditions.
From Eq. (7), the variation of the density of electrons
results in the spectral shift not only at the fundamental
wavelength of driving laser pulses, but also at the frequency of the harmonics. The former is determined as
the first term δλ0q δλ01 ∕q, whereas the latter by the second term δλ0q δλ01 ∕q3 . Since the term of the fundamental
pulses, 1∕q, is always larger than the term of harmonics
of 1∕q3 , the spectral modulation of the harmonics is dominated by the spectral shift of the driving laser pulses.
This is confirmed through the fitting of the spectral shift
of all the harmonic orders, in which we find that they are
fitted better with the factor of 1∕q than that of 1∕q3 .
At all the delay times around the quarter revival, the
observed spectral shifts for H27 are plotted in Fig. 3,
and we simulated the data by taking into account the free
electron effect, the molecular alignment effect, and the
(5)
where e is the electronic charge, m is the mass of the
electron, ε0 is the permittivity of vacuum, ω is the pulse
frequency, and N e denotes the electron density. The
spectral shift is therefore calculated by
δλ0 λL dδn0 λ3 e2 L dN e
λ3 e2 L ρ0 W t
≈− 2
− 2
;
3
c dt
8π ε0 mc dt
8π ε0 mc3 τon
(6)
where τon denotes the duration of the probe pulse. W t is
the ionization ratio at the delay time t when molecules
Fig. 3. Measured spectral shifts of H27 at all the time delays
(markers) around the quarter revival, and the simulation results
(solid lines) by taking into account (a) free electrons,
(b) molecular alignment excluding plasma effect, and (c) the
sum of the two effects, respectively. Square markers in (b)
are the experimental result minus plasma effect to highlight
the alignment contribution.
November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS
sum of the two effects, respectively. As one can see, the
measured Δλ27 roughly agrees with the effect of free electrons shown in Fig. 3(a), but cannot be simulated well
using only the molecular alignment effect. If the calculated contribution of the plasma is subtracted from the
data, the data can be simulated well by only considering
the molecular alignment effect as shown in Fig. 3(b),
which identifies the sole contribution of molecular alignment on the spectral shift of HHG. Figure 3(c) shows the
simulation result matching the experiment data better
when considering both the effects of free electrons
and molecular alignment.
In conclusion, spectral redshift and blueshift of HHG
have been observed at the modulated alignment conditions of CO2 molecules around the quarter revival. We
find that both the derivative of the alignment degree
and the density of free electrons result in the varied harmonic spectra. Analysis indicates that the effect of free
electrons on the refractive index on the fundamental laser pulses plays a dominant role in the modulation. The
spectral modulation of HHG from prealigned molecules
is experimentally identified for the first time. From the
finding one can fine tune the spectra of the harmonics
by selecting the alignment conditions of the molecules.
This work is supported by the National Natural
Science Foundation of China (Grant Nos. 11274326,
60978012, 11127901, and 11134010), National 973 Project
(Grant No. 2011CB808103), Chinese Academy of
Sciences and the State Key Laboratory of High Field
Laser Physics.
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