Backward surface electromagnetic waves in semi-infinite one-dimensional photonic crystals
containing left-handed materials
J. Barvestani,
1,2,
*
M. Kalafi,
1,2,3
A. Soltani-Vala,
1,2
and A. Namdar
4
1
Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz 51665-163, Iran
2
Physics Faculty, University of Tabriz, Tabriz 51665-163, Iran
3
Excellence Center for Photonics, University of Tabriz, Tabriz 51665-163, Iran
4
Physics Department, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran
Received 5 August 2007; published 11 January 2008
We study the electromagnetic surface waves localized at an interface separating a homogeneous dielectric
medium and a semi-infinite one-dimensional photonic crystal made of alternative left-handed metamaterial and
right-handed material. An analytical direct matching procedure within the Kronig-Penney model was applied to
analyze the dispersion properties of the localized surface states. We show that the presence of metamaterial in
the photonic crystal structure can support the surface waves with a backward energy flow and allows a flexible
control of dispersion properties of the surface modes. The surface states can be either forward or backward
waves depending on the physical parameters of the photonic crystal, physical parameters of the cap layer, the
position of the surface plane, and incident angle of the incoming beam.
DOI: 10.1103/PhysRevA.77.013805 PACS numbers: 42.70.Mp, 42.65.-k, 42.70.Qs, 73.20.At
I. INTRODUCTION
The left-handed metamaterial LHM with simultaneously
negative effective dielectric permittivity and effective mag-
netic permeability has recently attracted much attention due
to its unique physical properties and novel applications of
these materials 1,2 and triggered the debates on the appli-
cation of the left-handed slab as so-called “superlenses”
3,4. These materials can support an electromagnetic wave
where the phase propagation is antiparallel to the direction of
energy flow. Their properties were first considered theoreti-
cally by Veselago 5 during the 1960s but they have only
been fabricated recently 6,7. They are predicted to exhibit
many unusual properties such as refraction at a negative
angle, an inverse Doppler shift, and a backwards oriented
Chernekov radiation cone 5 negative giant Goos-Hanchen
effect 8. All these phenomena are rooted in the fact that the
phase velocity of light in LHM is opposite to the velocity of
energy flow, i.e., the Poynting vector and wave vector are
antiparallel so that the wave vector, the electric field, and the
magnetic field form a left-handed system.
Interfaces between different physical media can support a
special type of localized waves as surface waves or surface
modes, where the wave vector becomes complex causing the
wave to exponentially decay away from the surface. Surface
states have been studied in many different fields of physics,
including optics 9,10 where such waves are confined to an
interface between periodic and homogeneous dielectric me-
dia. In optics, the periodic structures have to be manufac-
tured artificially in order to manipulate dispersion properties
of light in a similar way as the properties of electrons are
controlled in crystals. Such periodic dielectric structures are
known as photonic crystals PC. An analogy between solid-
state physics and optics suggests that surface electromagnetic
waves should exist at the interfaces of photonic crystals, and
indeed they were predicted theoretically 9,10 and observed
experimentally 11. Such surface waves have some advan-
tages. First, surface states supported by PCs can exist in vir-
tually any optical frequency regime due to the scaling nature
of dielectric PCs. Second, the low dielectric loss in the struc-
tures can lead to sharp resonant coupling between the incom-
ing light and the surface states 12.
Band structure of one-dimensional 1D photonic crystals
containing alternative layers of left- and right-handed mate-
rials have been reported by Bria et al. 13. Recently it has
been shown that the interface separating a one-dimensional
conventional PC and a homogeneous left-handed material
can support backward Tamm states 14. The existence of
these modes depends only on the presence of homogeneous
left-handed materials. In this paper we have demonstrated
that the presence of metamaterials in the PC structure can
support backward surface modes localized at the interface
with right or left homogeneous medium. In this case, there is
more possibility to control the dispersion properties of sur-
face modes. In our model we study an electromagnetic sur-
face wave guided by an interface between right-handed
metamaterial RHM and a semi-infinite one-dimensional
photonic crystal consisting of alternate LHM and RHM lay-
ers which we refer to as L-R PC throughout this paper. We
assume that the terminating layer of the periodic structure
has the width the same or different from the width of other
layers of the structure. We study the effect of the width and
type of this termination layer on surface states and explore a
possibility to control the dispersion properties of surface
waves by adjusting termination layer thickness. We also
show that the presence of the LHM layers allows for a flex-
ible control of the dispersion properties of surface waves and
can support the unusual type of surface wave with a back-
ward energy flow.
In our calculations, the dielectric permittivity and mag-
netic permeability are, in general, assumed to take constant
values. Although these parameters in LHMs are in general
frequency dependent, our results can be used to design spe-
*
barvestani@tabrizu.ac.ir
PHYSICAL REVIEW A 77, 013805 2008
1050-2947/2008/771/0138055 ©2008 The American Physical Society 013805-1