Electromagnetic field of a moving charge in the presence of a left-handed medium
Sergey N. Galyamin* and Andrey V. Tyukhtin
†
Physics Department, St. Petersburg State University, St. Petersburg, 198504, Russia
Received 16 February 2010; revised manuscript received 5 May 2010; published 29 June 2010
We analyze the electromagnetic field of a moving point charge in the presence of a “left-handed” medium
LHM. First, the case of uniform motion of the charge in infinite LHM was considered. Using complex
function theory methods, we decomposed the total field into a “quasi-Coulomb” field, a wave field Cherenkov
radiation and a “plasma trace.” In addition, an effective numerical algorithm was developed for the total field
computation, and typical plots were presented. It was shown that the wave field in LHM lags behind the charge
more so than it does in ordinary medium. Furthermore, we investigated a charge intersecting the interface
between vacuumlike medium and LHM. Asymptotic expressions for the field components were obtained and
algorithms for their computation were developed. The spatial radiation can be separated into three distinct
components, corresponding to ordinary transition radiation with a relatively large magnitude, Cherenkov ra-
diation and reversed Cherenkov-transition radiation RCTR. Rigorous conditions for generating RCTR were
obtained: RCTR in the vacuum area was found to be the double threshold effect in both frequency and charge
velocity domain. The RCTR decay in vacuum due to the losses in LHM was studied. Areas of the RCTR
significance were determined and were found to be sufficiently large for observation. Possible applications to
the beam diagnostics and the characterization of metamaterials were suggested.
DOI: 10.1103/PhysRevB.81.235134 PACS numbers: 41.60.Bq, 41.60.Dk, 41.20.-q
I. INTRODUCTION
Recent progress in design and fabrication of metamateri-
als MTMs allows the realization of a variety of electromag-
netic properties not found in nature. MTM is, as a rule, a
periodic structure comprising certain conductive, dielectric,
or magnetic inclusions, with their size varying from hun-
dreds of nanometers to millimeters. For an electromagnetic
wave with a wavelength much larger than the unit-cell di-
mensions, the MTM is similar to a continuous medium and
can be described by effective macroscopic parameters
eff
and
eff
. One attractive example of “metamedia” with exotic
properties is that of “left-handed media” LHM.
The idea of LHM, i.e., media simultaneously having
negative permittivity and permeability , was introduced
in the 1960s by Veselago.
1,2
Contrary to ordinary “right-
handed” media RHM, vectors E
, H
, and k
form a left-
handed orthogonal set in LHM. Therefore, the Poyting vector
S
= c / 4E
, H
is opposite to the wave vector k
and to the
phase velocity. These peculiarities result in the unusual prop-
erties of wave processes in LHM.
1,2
In particular, an electro-
magnetic wave undergoes anomalous negative refraction at
the RHM-LHM interface: the tangential projection of the
Poyting vector of refracted wave is opposite to that of the
incident wave.
Until recently no materials with negative have been
found; therefore, LHM was merely a theoretical concept. At
the same time, long before Veselago’s papers,
1,2
certain ques-
tions concerning LHM were discussed indirectly by a num-
ber of researchers,
3–6
especially by Mandelshtam.
3,4
Man-
delshtam has demonstrated that the essence of negative
refraction is the “negative group velocity” effect, i.e., the
opposite orientation of the group velocity together with the
energy flow S
and the phase velocity.
Owing to the development of Pendry’s ideas,
7,8
LHM
were realized by means of MTMs.
9,10
The first real left-
handed MTM was composed of double split-ring resonators
and thin wires, producing negative
eff
and negative
eff
,
respectively, in a common range of gigahertz GHz
frequencies.
9,11
More recently MTMs with negative and
negative refractive index at frequencies starting from THz up
to the visible were demonstrated.
12–14
They use different unit
cell designs, including, for example, single split-ring resona-
tors, cut-wire pairs or “fishnet,” depending on the operating
frequency range.
15
Usually, the conductive elements are de-
posited at the dielectric substrate to get one monolayer of
MTM. Distributing monolayers in an appropriate manner in
the space, the three-dimensional structure can be built. It is
remarkable that negative in natural medium ferromagnetic
metal was also observed recently,
16
expanding the possibili-
ties for the realization of LHM.
The “left-handed properties” can, in principle, be realized
only inside a limited frequency range.
1,2
Therefore, it would
be more correct to refer to a “left-handed frequency band”
LHFB and a “right-handed frequency band” RHFB in-
stead of LHM and RHM. However, as the term LHM is
widely used in scientific literature, we will use it as well with
the understanding that LHM is a medium possessing both
LHFB and RHFB whereas RHM is a medium with RHFB
only.
Starting from the first realization, different electromag-
netic processes in LHM are actively investigated. Papers deal
with the plane waves propagation in LHM and with their
refraction at the RHM-LHM interface,
11,17–20
as well as with
the extraction of the effective parameters
eff
and
eff
from
scattering data.
21–23
Several papers are devoted to theoretical
analysis of Cherenkov radiation CR in infinite or semi-
infinite LHM Refs. 24 –26in particular, the effect of re-
versed CR was mentioned
24
or to design and fabrication of
the specific type of MTM suitable for the CR observation.
27
A recent experiment has proven the reversed nature of CR in
LHM.
28
Transition radiation TR at the boundary with LHM
was also partially analyzed.
29,30
Pafomov
29
appears to be the
PHYSICAL REVIEW B 81, 235134 2010
1098-0121/2010/8123/23513414 ©2010 The American Physical Society 235134-1