Physics Letters A 362 (2007) 1–5
www.elsevier.com/locate/pla
Generation of harmonics by a focused laser beam in the vacuum
A.M. Fedotov, N.B. Narozhny
∗
Moscow Engineering Physics Institute, 115409 Moscow, Russia
Received 18 September 2006; accepted 22 September 2006
Available online 5 October 2006
Communicated by V.M. Agranovich
Abstract
We consider generation of odd harmonics by a super strong focused laser beam in the vacuum. The process occurs due to the plural light-
by-light scattering effect. In the leading order of perturbation theory, generation of (2k + 1)th harmonic is described by a loop diagram with
(2k + 2) external incoming, and two outgoing legs. A frequency of the beam is assumed to be much smaller than the Compton frequency, so
that the approximation of a constant uniform electromagnetic field is valid locally. Analytical expressions for angular distribution of generated
photons, as well as for their total emission rate are obtained in the leading order of perturbation theory. Influence of higher-order diagrams is
studied numerically using the formalism of Intense Field QED. It is shown that the process may become observable for the beam intensity of the
order of 10
27
W/cm
2
.
©
2006 Elsevier B.V. All rights reserved.
PACS : 12.20.Ds; 11.80.La; 41.85.-p; 42.65.Ky
Keywords: Super strong laser field; Focused beam; Harmonics generation; Stimulated emission
The latest achievements in laser technology promise very
rapid growth of peak laser intensities. In a recent paper [1]
Tajima and Mourou suggested a path to reach an extremely high
intensity level 10
26
–
28
W/cm
2
already in the coming decade,
taking advantage of the megajoule facilities, see also Refs. [2,
3]. The field strength for such lasers will be very close to the
characteristic QED value E
S
= m
2
c
3
/e
¯
h= 1.32× 10
16
V/cm,
and thus nonlinear QED vacuum polarization effects will be-
come measurable. Some of them have been already studied
in literature. Among those, harmonic generation by an intense
laser beam propagating in an external magnetic field [4], and
by two colliding beams in the vacuum [5]. Recently the experi-
mental feasibility of the light-by-light stimulated scattering via
three laser beams has been studied in Ref. [6]. Pair creation by
a focused laser pulse in the vacuum was considered in Ref. [7],
and by two colliding pulses in Ref. [8]. Other references could
be found in the recent review [9]. In this Letter, we consider the
*
Corresponding author.
E-mail addresses: fedotov@cea.ru (A.M. Fedotov),
narozhny@theor.mephi.ru (N.B. Narozhny).
effect of odd harmonics generation by a strong focused laser
beam in the vacuum. Clearly, the probability of this process
is smaller compared to an analogous light-by-light scattering
effect in a combination of laser beams [6]. However it is of
great importance, since the superhigh laser intensities can be
achieved only in focused and very short laser pulses, and hence
the harmonics generation effect will necessarily be present at
any facility producing pulses with the peak field strength of the
order of E
S
.
The process under consideration is described by the Feyn-
man diagram depicted in Fig. 1(a). In the two leading orders
of perturbation theory the effect is represented by diagrams
in Fig. 1(b), (c). The diagrams without outgoing laser pho-
tons do not contribute since they would correspond to creation
of an extraneous photon by a group of laser photons propa-
gating in one direction. As is well known, such photons do
not interact. For the fields weakly varying at distances of the
order of the Compton length l
C
, and time intervals l
C
/c, non-
linear vacuum polarization effects can be described using the
Heisenberg–Euler radiative correction to the electromagnetic
field Lagrangian density [10–12], which can be represented by
0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.physleta.2006.09.085