Harmonic generation by atoms in circularly polarized two-color laser fields with coplanar polarizations and commensurate frequencies

Abstract

The generation of harmonics by atoms or ions in a two-color, coplanar field configuration with commensurate frequencies is investigated through both, an analytical calculation based on the Lewenstein model and the numerical ab initio solution of the time-dependent Schroedinger equation of a two-dimensional model ion. Through the analytical model, selection rules for the harmonic orders in this field configuration, a generalized cut-off for the harmonic spectra, and an integral expression for the harmonic dipole strength is provided. The numerical results are employed to test the predictions of the analytical model. The scaling of the cut-off as a function of both, one of the laser intensities and frequency ratio $\eta$, as well as entire spectra for different $\eta$ and laser intensities are presented and analyzed. The theoretical cut-off is found to be an upper limit for the numerical results. Other discrepancies between analytical model and numerical results are clarified by taking into account the probabilities of the absorption processes involved. Comment: 8 figures

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arXiv:physics/0211110v1 [physics.atom-ph] 27 Nov 2002
Harmonic generation by atoms in circularly polarized two-color
laser fields with coplanar polarizations and commensurate
frequencies
F. Ceccherini,1, ∗D. Bauer,2and F. Cornolti1
1Istituto Nazionale per la Fisica della Materia (INFM),
sez. A, Dipartimento di Fisica “Enrico Fermi”,
Universit` a di Pisa, Via F. Buonarroti 2, 56127 Pisa, Italy
2Theoretical Quantum Physics (TQP),
Darmstadt University of Technology,
Hochschulstr. 4A, D-64289 Darmstadt, Germany
(Dated: February 2, 2008)
Abstract
The generation of harmonics by atoms or ions in a two-color, coplanar field configuration with
commensurate frequencies is investigated through both, an analytical calculation based on the
Lewenstein model and the numerical ab initio solution of the time-dependent Schr¨ odinger equation
of a two-dimensional model ion. Through the analytical model, selection rules for the harmonic
orders in this field configuration, a generalized cut-off for the harmonic spectra, and an integral
expression for the harmonic dipole strength is provided. The numerical results are employed to test
the predictions of the analytical model. The scaling of the cut-off as a function of both, one of the
laser intensities and frequency ratio η, as well as entire spectra for different η and laser intensities
are presented and analyzed. The theoretical cut-off is found to be an upper limit for the numerical
results. Other discrepancies between analytical model and numerical results are clarified by taking
into account the probabilities of the absorption processes involved.
PACS numbers: 42.50.Hz, 42.65.Ky
∗Electronic address: ceccherini@df.unipi.it
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I.INTRODUCTION
The possibility of obtaining high frequency radiation through the interaction of a laser
field and an atom is a topic that has been extensively addressed during the last two decades
from both a theoretical and an experimental point of view (see [1] for recent reviews).
Harmonic generation from sources other than atoms, like linear molecules [2], ring molecules
(e.g. benzene) [3, 4], nanotubes [5], and plasmas [6] have been also investigated.
From the invariance of the Hamiltonian under dynamical symmetry operations, selection
rules for harmonic generation can be elegantly derived [3, 7]. It turns out that apparently
very different target and field configurations yield selection rules of the same type. Let us
consider, e.g., a ring molecule with N ions (e.g., N = 6 in the case of benzene) in a circularly
polarized laser field of frequency ω. The electric field vector lies in the plane that is spanned
by the molecule and which we parameterize through the polar coordinates ρ and ϕ. The
corresponding Hamiltonian (with the laser interaction taken in dipole approximation) is
invariant under the dynamical symmetry operation
ˆPN=
?
ρ → ρ,ϕ → ϕ +2π
N,t →
2π
Nω
?
.(1)
From this invariance follows [3, 7] that only harmonics of order
n = gN ± 1,g ∈ N+
(2)
can be emitted. The harmonic radiation is circularly polarized and subsequent harmonics
are alternately clockwise and counter-clockwise polarized. More complicated selection rules
arise when also excited states are taken into account [7]. Let us now turn to the actual
target and field configuration examined in the present paper, namely the situation of an
atom (or ion) in a circularly polarized two-color laser field of frequencies ω and ηω with η a
positive integer number, and coplanar polarizations. In the case of counter-rotating electric
field vectors the Hamiltonian is invariant under the same symmetry operation (1) with N
replaced by η +1. The polar coordinates ρ and ϕ are with respect to the polarization plane
now. In the case of co-rotating electric field vectors N has to be replaced by η − 1.
An appealing feature of the selection rule (2) is the fact that with increasing N, i.e.,
number of ions or frequency ratio of the two laser fields, respectively, less harmonics are
emitted within a fixed frequency interval. This filtering effect may be accompanied with more
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efficient emission of harmonics at short wavelengths which are of interest in spectroscopic
applications, for instance.
Harmonic generation in two-color fields has been studied both experimentally [8] and
theoretically [9, 10, 11].
The present paper is organized as follows: In Sec. II the theoretical modeling proposed
by Lewenstein et al. [12] for harmonic generation in the case of an atom interacting with a
single linearly polarized field is extended to the two-color configuration, and several expected
features of the harmonic spectra are deduced, among them the scaling of the cut-off and the
dependence of the relative dipole strengths within a certain harmonic couple g. In Sec. III ab
initio numerical results obtained through the integration of the time-dependent Schr¨ odinger
equation for a two-dimensional model ion are presented and compared with the predictions
by the analytical model. Finally, a conclusion is given in Sec. IV.
Atomic units (a.u.) are used throughout the paper.
II. ANALYTICAL THEORY
A theory of harmonic generation should answer mainly two fundamental questions: (i)
which harmonics are emitted and (ii) which is the intensity of the harmonics as a function
of the laser and target parameters. These questions have been addressed in [12] for the
case of an atom (in the single-electron approximation) interacting with a linearly polarized
laser field (in dipole approximation). In a similar approach, the more general and more
complicated case of elliptical polarization was studied in [13] (single color). The elliptically
polarized two-color field was addressed in [14] but the discussion of the cut-off law as well as
the presentation of the numerical results were restricted to linear polarization there. Here, we
focus on the case of two laser fields with circular polarizations and arbitrary integer frequency
ratios and compare carefully the model predictions with ab initio numerical simulations.
The electric field caused by the two lasers of frequency ω1= ω and ω2= ηω is assumed
to be
?E(t) = (Ex(t),Ey(t),Ez(t)) = (E1cos(ωt) + E2cos(ηωt),E1sin(ωt) − E2sin(ηωt),0)(3)
where E1and E2are the amplitude of the first and the second laser field, respectively, and
the dipole approximation is applied. The two fields are oppositely polarized and coplanar.
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The case of co-rotating electric field vectors will be discussed later-on. The vector potential
?A(t) = −?tE(t′)dt′reads
?A(t) = (Ax(t),Ay(t),Az(t)) = −
?E1
ωsin(ωt) +E2
ηωsin(ηωt),−E1
ωcos(ωt) +E2
ηωcos(ηωt),0
?
.
(4)
Our starting point is the dipole moment along the direction ? n as it is calculated in the
Lewenstein model (cf. Eq. (10) of Ref. [12])
x? n(t) = i
?t
0dt′
?
d3p? n ·?d∗(? p −?A(t))?E(t′) ·?d(? p −?A(t′))exp[−iS(? p,t,t′)] + c.c..(5)
Here,
?d(? p) = ?? p|? r|0?,(6)
and the action S(? p,t,t′) is given by
S(? p,t,t′) =
?t
t′

(? p −?A(t′′))2
2
+ Ip

dt′′.(7)
In order to arrive at expression (5) several assumptions have been made in [12]: (i) among
the bound states only the ground state plays a role in the evolution of the system; (ii) the
depletion of the ground state can be neglected; (iii) in the continuum V (? r) plays no role and
the electron is treated like a free particle and can be therefore described py plane waves |? p?;
and (iv) contributions from continuum-continuum transitions to harmonic generation can
be neglected.
For the field (3), (4), the general expression (5) evaluated for, e.g., ? n = ? ex, reads
x(t) = i
?t
0dt′
?
d3p
?
Ex(t′)dx(? p −?A(t′)) + Ey(t′)dy(? p −?A(t′)
?
d∗
x(? p −?A(t))exp[−iS(? p,t,t′)]
(8)
(the “+ c.c.” is suppressed from now on).
The integration over ? p is performed approximately by means of the saddle-point method,
assuming that the major contribution to the integral is given by stationary points of the
classical action, i.e, the points pst
x, pst
ythat satisfy
?∇? pS(? p,t,t′) =?0.(9)
One finds
pst
x(t,τ) =
E1
ω2τ
?
cos(ωt) − cos(ω(t − τ))
?
+
E2
η2ω2τ
?
cos(ηωt) − cos(ηω(t − τ))
?
,(10)
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pst
y(t,τ) =
E1
ω2τ
?
sin(ωt) − sin(ω(t − τ))
?
−
E2
η2ω2τ
?
sin(ηωt) − sin(ηω(t − τ))
?
(11)
where τ = t − t′is the electron’s “travel time.”Introducing the stationary action
Sst(t,τ) = S(? pst,t,t − τ) =
?t
t−τ

(? pst(t,τ) −?A(t′′))2
2
+ Ip

dt′′
(12)
with ? pst= (pst
x,pst
y,0) we obtain, after the saddle-point integration over ? p,
x(t) = i
?∞
?
0
dτ
?
π
ǫ + iτ/2
?3/2
exp(−iSst(t,τ))d∗
x(? pst(t,τ) −?A(t))
×
Ex(t − τ)dx(? pst(t,τ) −?A(t − τ)) + Ey(t − τ)dy(? pst(t,τ) −?A(t − τ))
?
. (13)
The factor with infinitesimal ǫ in (13) comes from the regularized Gaussian integration over
? p. It expresses quantum diffusion of the released wave packet and damps away contributions
from times τ much larger than a laser cycle, allowing for the extension of the τ-integration
to infinity [12].
The stationary action (12) can be written in the form
Sst(t,τ) = C0(τ) + C1(τ)cos((η + 1)ω(t − τ/2)) (14)
where C0(τ) and C1(τ) are given by
C0(τ) = Ipτ +
E2
2ω4τ(τ2ω2− 2 + 2cos(τω)) +
4E1E2
η2ω4τsin(ωτ/2)sin(ηωτ/2) −
1
E2
2
2η4ω4τ(η2τ2ω2− 2 + 2cos(ητω)), (15)
2E1E2
η(η + 1)ω3sin((η + 1)ωτ/2).
C1(τ) =
(16)
The expression (14) for the quasi-classical action is very useful and interesting. The time
dependence is given by just one term and through only one effective frequency which is
(η +1)ω. This is consistent with the selection rule g(η +1)±1 obtained previously. Taking
η = 1 and E1= E2= E/2 the coefficients C0(τ) and C1(τ) calculated in [12] for the single
linearly polarized field are easily recovered.
As the semi-classical action is the integral over time of the kinetic energy, an expression
for the energy gain of the electron is obtained by deriving Sst(t,τ) with respect to t,
∆Ekin(t,τ) = Ekin(t) − Ekin(t − τ) =∂Sst(t,τ)
∂t
= −(η + 1)ωC1(τ)sin
?
(η + 1)ω
?2t − τ
2
??
.(17)
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