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Dependence of high-order-harmonic-generation yield on driving-Dependence of high-order-harmonic-generation yield on driving-
laser ellipticity laser ellipticity
Max Möller
University of Central Florida
Yan Cheng
University of Central Florida
Sabih D. Khan
Baozhen Zhao
Kun Zhao
University of Central Florida
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Recommended Citation Recommended Citation
Möller, Max; Cheng, Yan; Khan, Sabih D.; Zhao, Baozhen; Zhao, Kun; Chini, Michael; Paulus, Gerhard G.; and
Chang, Zenghu, "Dependence of high-order-harmonic-generation yield on driving-laser ellipticity" (2012).
Faculty Bibliography 2010s
. 3041.
https://stars.library.ucf.edu/facultybib2010/3041
RAPID COMMUNICATIONS
PHYSICAL REVIEW A 86, 011401(R) (2012)
Dependence of high-order-harmonic-generation yield on driving-laser ellipticity
Max M
¨
oller,
1,2,3
Yan Cheng,
3
Sabih D. Khan,
4
Baozhen Zhao,
5
Kun Zhao,
3
Michael Chini,
3
Gerhard G. Paulus,
1,2,6
and Zenghu Chang
3,*
1
Institut f
¨
ur Optik und Quantenelektronik, Friedrich-Schiller Universit
¨
at Jena, 07743 Jena, Germany
2
Helmholtz Institut Jena, Helmholtzweg 4, 07743 Jena, Germany
3
CREOL and Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
4
Department of Physics, Kansas State University, Manhattan, Kansas 66506, USA
5
Department of Physics and Astronomy, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
6
Department of Physics, Texas A&M University, College Station, Texas 77843, USA
(Received 23 April 2012; published 26 July 2012)
High-order-harmonic-generation yield is remarkably sensitive to driving laser ellipticity, which is interesting
from a fundamental point of view as well as for applications. The most well-known example is the generation
of isolated attosecond pulses via polarization gating. We develop an intuitive semiclassical model that makes
use of the recently measured initial transverse momentum of tunneling i onization. The model is able to predict
the dependence of the high-order-harmonic yield on driving laser ellipticity and is in good agreement with
experimental results and predictions from a numerically solved time-dependent Schr
¨
odinger equation.
DOI: 10.1103/PhysRevA.86.011401 PACS number(s): 32.80.Wr, 42.65.−k, 33.20.Xx
High-order-harmonic generation (HHG) in gases is
presently the most important method for generating extreme
ultraviolet (XUV) attosecond pulses from intense infrared
lasers [1,2]. The semiclassical picture of HHG divides the
process into three steps that take place within less than an
optical cycle [3,4]. First, the single active electron is ionized
from the bound state t o the continuum. During the second
step, the electron travels in the continuum influenced by the
laser field. The XUV photon emission takes place in the
third step when the electron is driven back to the parent ion
and recombines about half an optical cycle after the initial
ionization. The ionization step is fundamental for the behavior
of the entire HHG process. Typically, it is treated by tunneling
of a bound single active electron through a quasistatic potential
barrier arising from the superposition of the Coulomb field and
the strong laser field [5]. This model of tunneling at optical
frequencies has been under experimental [6] and theoretical
investigation over the past years [7,8]. Here, we develop
a semiclassical model that qualitatively and quantitatively
explains the dependence of high-order-harmonic-generation
yield on driving laser ellipticity based on the transverse
velocity distribution of the electron wave packet at the exit
of the tunnel. The model utilizes classical mechanics to reveal
the subcycle electron dynamics in an elliptically polarized laser
field [9].
Our results can be used to design and further optimize
techniques to generate isolated attosecond laser pulses from
multicycle driving lasers which emit attosecond XUV pulses
each half cycle of the driving laser, i.e., separated by only about
1 fs in time, unless specific technical measures are taken. An
obvious way for generating isolated attosecond pulses is the
use of driving laser pulses consisting of essentially a single
optical cycle [10]. The construction of such quasi-single-cycle
lasers is quite demanding [11]. This holds in particular if
*
To whom correspondence should be addressed:
zenghu.chang@ucf.edu
a high driving laser pulse energy (>100 mJ), which has a
positive impact on HHG flux, is a prime design criterion. An
alternative approach to generate isolated attosecond pulses is
to manipulate conventional multicycle laser pulses such that
only a single optical cycle can contribute to HHG. The basis for
most of such methods is the remarkable sensitivity of HHG
efficiency on driving laser ellipticity. Thus, by tailoring the
polarization such that only a single optical cycle contributes
to HHG, it is also possible to produce isolated attosecond
pulses. This technique is known as polarization gating (PG)
[12,13]. Since its proposal in 1994 [14], polarization gating
has become a frequently used technique. In fact, a variety
of innovations for such gating methods have expanded their
range of applicability. Examples are interferometric PG [15],
double optical gating [16,17], and generalized double optical
gating [18](seeRef.[19]forareview).
However, the decisive physical effect is still the depen-
dence of high-order-harmonic yield on driving laser elliptic-
ity [20,21]. Our semiclassical model identifies the physical
mechanism of the ellipticity dependence in HHG. A parametric
study of the effect performed at 810 and 405 nm driving laser
wavelength as well as experimental and theoretical results
from the literature are found to be in good agreement with
the model. As the analysis allows calculating the ellipticity
dependence of the yield in HHG as a function of the driving
laser wavelength, intensity, target atom, and harmonic order, it
should be very useful for designing polarization gating-based
schemes for the generation of isolated attosecond pulses from
multicycle driving lasers.
As briefly introduced, the semiclassical theory of HHG is
based on the analysis of classical electron trajectories that
evolve when atoms release electrons in a strong oscillating
electric field. Of particular interest are those trajectories that
return to the parent ion core. Since the kinetic energy of
the returning electron can exceed the photon energy of the
driving laser field by orders of magnitude, recombination
will lead to the emission of extreme ultraviolet photons. This
already suggests the explanation for the strong dependence of
011401-1
1050-2947/2012/86(1)/011401(5) ©2012 American Physical Society
RAPID COMMUNICATIONS
MAX M
¨
OLLER et al. PHYSICAL REVIEW A 86, 011401(R) (2012)
high-order-harmonic yield on driving laser ellipticity. One of
the transverse components of the elliptically polarized field
will prevent the electrons from returning to the ion core and
thus switch off the mechanism of HHG.
For a quantitative analysis, quantum features of the ion-
ization process have to be taken into account. The respective
semiclassical model of the ellipticity dependence of HHG is
based on the assumption that the HHG radiation is due to
electron trajectories where the transverse displacement caused
by the external field is compensated by an initial transverse
velocity of the electron at the exit of the tunnel. Electrons
following these trajectories (which lead exactly back to the
ion core) have a higher probability of recombination with the
parent ion and emission of an HHG photon than trajectories
missing the ion core.
The trajectories favorable for HHG can be found easily
when the Coulomb field of the ion is neglected. Integrating the
equations of motion for a free electron in an elliptically polar-
ized laser field t hat is approximated as a monochromatic plane
wave,
F (t) = F/
√
1 + ε
2
[cos ωt; ε sin ωt], with ellipticity ε,
amplitude F , and frequency ω, yields the trajectory,
r =−
F
√
1 + ε
2
1
ω
2
×
−cos ωt + cos ωt
0
− ω
(
t −t
0
)
sin ωt
0
ε
(
−sin ωt + sin ωt
0
+ ω
(
t −t
0
)
cos ωt
0
)
+
(
t −t
0
)
v
x0
v
y0
. (1)
Here, in the spirit of the semiclassical strong-field model,
the initial positions x
0
and y
0
are set to zero and t
0
denotes the
time of ionization. Atomic units are used except where noted.
The angle between the ionizing field vector and the
coordinate axis at the time of ionization is given by α
0
=
tan
−1
[
ε tan(ωt
0
)
]
. A rotation R(−α
0
) is applied to express the
initial velocities v
x0
and v
y0
in terms of parallel and transverse
velocities to the ionizing field vector, v
and v
⊥
,
v
x0
v
y0
= R(−α
0
)
v
v
⊥
=
v
cos(α
0
) + v
⊥
sin(α
0
)
−v
sin(α
0
) + v
⊥
cos(α
0
)
. (2)
As the electron leaves the atom by tunneling, we assume
that the initial velocity parallel to the ionizing field is zero,
v
= 0.
For nonzero ellipticity ε and zero initial velocity v
=
v
⊥
= 0, the electron can never return to the parent ion at
the origin. Consequently, high-order-harmonic emission is
avoided, as illustrated in Fig. 1(a). However, a return at a later
recombination time, t
r
>t
0
, can be achieved if the trajectory
is launched with a finite initial velocity v
⊥
perpendicular
to the field direction at t = t
0
. This idea was stressed, e.g.,
in the trajectory analysis for laser fields with a polarization
gate [22,23]. For a given starting time, the return time t
r
can
be found by solving the condition of return r
(
t
r
)
= 0, which
yields [24]
(ε
2
+1)ω(t
r
−t
0
)sinωt
0
cos ωt
0
−ε
2
sin ωt
0
(sin ωt
r
−sin ωt
0
)
+ cos ωt
0
(cos ωt
r
− cos ωt
0
) = 0. (3)
-140 -120 -100 -80 -60 -40 -20 0
0
20
40
60
80
100
120
y [a.u.]
x [a.u.]
400 nm
800 nm
1600 nm
⊥
v
-1.0 -0.5 0.0 0.5 1.0
1
Probability
Transverse Velocity [a.u.]
Neon
-140 -120 -100 -80 -60 -40 -20
02
02
0
-300
-250
-200
-150
-100
-50
0
I = 3.5x10
14
ε
= 0.5
t
0
= 0.05 T
y [a.u.]
(a)
(b)
FIG. 1. (Color online) (a) Electron trajectories in elliptically
polarized fields driven at different wavelengths. The trajectories start
at the instant of ionization, here t
0
= 0.05T , and terminate when
the electron crosses the x axis. The movement along the minor field
component causes a displacement which easily reaches tens of atomic
units and is proportional to the square of the wavelength, ∝ λ
2
.
(b) Electron trajectories where an initial transverse velocity v
⊥
(indicated by the arrows) compensates the displacement, thus fa-
cilitating the return of the trajectory to the origin. The inset shows the
probability distribution for an initial transverse velocity for neon at
an intensity of 3.5 ×10
14
W/cm
2
.
The required initial velocity v
⊥
then is given by
v
⊥
=−
Fε
√
1 + ε
2
1
ω cos α
0
sin ωt
r
− sin ωt
0
ω
(
t
r
− t
0
)
− cos ωt
0
.
(4)
The resulting trajectory with the initial perpendicular
velocity is adjusted such that the trajectory immediately returns
to the ion core, as shown in Fig. 1(b).
Although a suitable transverse initial electron velocity
facilitates the collision of the returning electron with the ion
and thus increases the probability of recombination, one also
has to take into account that electrons leaving t he tunnel are
unlikely to have large transverse velocities [5–8]. This changes
the probability of finding such an electron trajectory, as can
be seen from the inset in Fig. 1(b) . The ellipticity-dependent
yield of high-order harmonics normalized to the yield with a
linearly polarized laser is therefore
I
XUV
(ε)
I
XUV
(ε = 0)
≈
w
(
v
⊥
)
w
(
v
⊥
= 0
)
= exp
−
2I
p
v
⊥
(ε)
2
|
F (t
0
)
|
, (5)
011401-2
RAPID COMMUNICATIONS
DEPENDENCE OF HIGH-ORDER-HARMONIC-GENERATION ... PHYSICAL REVIEW A 86, 011401(R) (2012)
where w(v
⊥
) is the perpendicular velocity distribution that
was predicted by tunneling theory [8] and found to agree well
with experimental results [6]. Here, I
p
denotes the ionization
potential and
|
F (t
0
)
|
is the absolute value of the driving field
at the instant of ionization.
In order to calculate the yield of high-order-harmonic
radiation at a given ellipticity, one identifies t
r
for all t
0
in
the first quarter of the optical cycle (0 <t
0
<T/4) by solving
Eq. (3) numerically, considering only the first quarter cycle is
sufficient due to the driving laser field’s periodicity. In fact,
Eq. (3) has in general several solutions t
r
which are related to
different v
⊥
by Eq. (4). For HHG, however, only the first
return of the electron to the ion is relevant, i.e., only t he
solution with the smallest t
r
needs to be taken into account.
According to the classical model of HHG, the photon energy
that is emitted by a trajectory starting at t
0
is given by the
electron’s kinetic energy upon its return at t
r
plus the ionization
energy of the target atom I
p
. Similar t o the case of linear
polarization, there are two types of trajectories that lead to
identical photon energy but have a different starting time t
0
,
return time t
r
, and initial velocity v
⊥
[25]. In the literature,
they are known as short and long trajectories. It is also known
that harmonics generated by long trajectories have undesirable
phase-matching properties, which is why experimental condi-
tions are usually chosen such that only the short trajectories
contribute to the observed harmonics. Thus, we calculate
the ellipticity-dependent yield using Eq. (5) for the short
trajectories.
An analytical expression for the ellipticity-dependent yield
can be derived based on the following assumptions: (i) A
small ellipticity (ε
2
1) which results in
√
1 + ε
2
≈ 1. This
is reasonable as the yield drops quickly as ε is increased,
particularly at longer driving laser wavelengths. (ii) Photon
energies that are close to the cutoff are related to starting
times shortly after the field reached a maximum. Therefore,
one can neglect the rotation of the field vector at the instant
of ionization, cos α
0
= cos{tan
−1
[ε tan(ωt
0
)]}≈1, in Eq. (4).
With the same argument, one approximates
|
F (t
0
)
|
≈
√
I in
Eq. (5), where I is the peak intensity of the field. (iii) Using
(i) and (ii) one can write Eq. (4) as v
⊥
=−βFελ/2πc, where
β = (
sin ωt
r
−sin ωt
0
ω(t
r
−t
0
)
− cos ωt
0
). c is the speed of light and λ
denotes the laser wavelength. β depends on the harmonic
order of interest and within the approximations (i) and (ii),
it can be calculated using closed-form analytical solutions for
linear polarization [26].
The resulting ellipticity-dependent yield is approximately
a Gaussian function which results directly from the velocity
spreading of the electron wave packet at the exit of the
tunnel,
I
XUV
(ε)
I
XUV
(ε = 0)
≈ exp
−
β
2
2I
p
I
4π
2
c
2
λ
2
ε
2
. (6)
The Gaussian dependence of high-order-harmonic yield on
ellipticity ε reflects the Gaussian dependence of the ionization
probability on v
⊥
. This emphasizes that high-order-harmonic
radiation from an elliptical polarized laser field comes from
electron trajectories where the transverse displacement is
compensated by an initial transverse velocity.
For the design of optical gating schemes, the threshold
ellipticity ε
th
is of particular interest as this quantity determines
the amplitude of satellite pulses that arise from radiation
generated outside the polarization gate [12]. Here, we choose
ε
th
as the ellipticity where the normalized yield drops to 0.1.
For cutoff harmonics, where β
2
≈ 1.59, one finds
ε
th
≈
691
I
1/4
p
1
I
1/4
1
λ
. (7)
Equation (7) shows a 1/λ scaling as was found exper-
imentally [27] and theoretically based on time-dependent
Schr
¨
odinger equation (TDSE) simulations [28].
The predictions of the model are compared with measure-
ments of the ellipticity-dependent high-order-harmonic yield
in using 405- and 810-nm driving lasers. We used the setup
that was presented in Ref. [29]. The 810-nm, 30-fs, 4.8-mJ
pulses were frequency doubled in a 300-μm beta barium borate
crystal (type I phase matching) to produce 0.9-mJ pulses of
second harmonic radiation centered at 405 nm. Two dichroic
mirrors removed the fundamental radiation. The ellipticity
was controlled by a combination of a zero-order half-wave
plate and a quarter-wave plate such that the orientation of
the polarization ellipse did not change as the ellipticity was
changed. This minimizes the influence of the polarization-
dependent diffraction efficiency of the toroidal grating in the
XUV spectrometer. The pulses were focused into a 1-mm-long
gas cell using a 375-mm silver-coated concave mirror at near
normal incidence. The fundamental radiation was blocked by
a 300-nm aluminum filter before the XUV spectrometer. The
ellipticity-dependent yield from 810-nm driving laser pulses
was measured using the same setup but with optics that were
designed for 810 nm.
In Fig. 2, experimental results from neon and helium
are compared with corresponding theoretical results. The
numerical results match the experiments quite well, while the
analytical results show the qualitative trend but larger quantita-
tive deviations for plateau harmonic orders far from the cutoff.
This is expected due to the breakdown of the aforementioned
assumptions. In particular, neglecting effects due the rotated
field vector at t
0
and assuming starting times that are close
to the maximum of the field fail for low-energy plateau
harmonics.
To further validate the model, the threshold ellipticities ε
th
of different harmonic orders at 810 and 405 nm are examined
in Figs. 3(a) and 3(b). The theory matches the trend that
higher harmonic orders are more susceptible to the ellipticity
and therefore have lowered threshold ellipticity. This finding
has implications for the design of polarization-based gating
schemes.
The threshold ellipticity of the 11th harmonic of 405 nm
as a function of i ntensity is shown i n Fig. 3(c). Theory and
experiment show that higher intensity results in a narrower
ellipticity dependence and thus a smaller threshold ellipticity.
Higher intensity causes a larger excursion amplitude of the
trajectory and therefore requires a larger initial transverse
velocity for the trajectory to return. However, the width of the
transverse velocity distribution is also increased, as evident
from Eq. (5). This mutual compensation explains why the
intensity has a relatively weak influence on the threshold
ellipticity.
011401-3
