Full-wave analysis of imaging by the Pendry-Ramakrishna stackable lens
A. V. Dorofeenko,
1
A. A. Lisyansky,
2
A. M. Merzlikin,
1
and A. P. Vinogradov
1,
*
1
Institute of Theoretical and Applied Electromagnetism, Russian Academy of Sciences, 125412 Moscow, Izhorskaya, 13/19, Russia
2
Department of Physics, Queens College of the City University of New York, Flushing, New York 11367, USA
Received 10 April 2006; published 30 June 2006
We perform a full-wave analysis of a stackable lens proposed in a recent paper Ramakrishna et al., J. Mod.
Optics 50, 1419 2003. This lens was suggested for improving subwavelength imaging and can be obtained
by splitting a single-layer lens into a set of thinner layers. Our analysis shows that i such a lens, which forms
a one-dimensional photonic crystal PC, is a resonator cavity for traveling Bloch waves that cannot leave this
PC resonator due to total internal reflection; ii imaging is possible outside the band gaps only and no imaging
can be achieved in the vicinity of the eigenstates of the PC resonator as well as near the state associated with
the excitation of the volume plasmon; iii the expected advantage is due to thinning the layers, which results
in shifting of both the band edge and the eigenstates toward higher values of the wave number; and iv a
single-layer lens has the broadest working range compared to a stackable lens with the same elementary layer
thickness.
DOI: 10.1103/PhysRevB.73.235126 PACS numbers: 78.20.Ci, 42.30.Wb, 42.70.Qs, 73.20.Mf
I. INTRODUCTION
In 1968, Veselago
1
showed that a medium with both nega-
tive permittivity and permeability a left-handed medium
exhibited a negative refraction as predicted in Ref. 2. The
most pronounced property of the left-handed medium, pre-
dicted by Veselago, is that a slab made of such a material
acts as a focusing lens, producing a real image of a source
placed in front of the slab. In 2000 Pendry
3
showed that an
image produced by the Veselago lens consists not only of far
field harmonics but also of near field harmonics of the
source. Thus, it has become possible to overcome the Ray-
leigh limit in imaging employing surface plasmon-magnon
resonances excited by the near fields.
3,4
As a result, a slab of
thickness L at a distance l
2
behind the lens perfectly restores
the image of the source placed at a distance l
1
in front of the
lens, where these distances are related by the equation l
1
+ l
2
= L.
3
The latter fact is very important in photolithography
because the left-handed slab shifts the perfect image from a
stencil-photoresist interface to deeper inside the photoresist,
making the performance of the ultimate mask better.
5,6
However, a left-handed medium is still not known in op-
tics; therefore, Pendry
3
suggested using a medium e.g., sil-
ver with negative permittivity only for optical applications.
Such a medium does not exhibit negative refraction, and
space harmonics with small values of the transverse wave
number k
x
are not focused and do not create an image. Nev-
ertheless, in a slab with
2
=-1,
2
= 1 placed in the xy plane,
the TM space harmonics with high values of k
x
are still am-
plified by the plasmon resonance, which makes the resolu-
tion of small details possible.
To describe wave propagation through such a slab for
high values of k
x
k
0
/ k
x
1, k
0
= / c, one can use an elec-
trostatic approximation.
3
In the Maxwell equations, one can
ignore the time derivatives compared to the spatial ones. In
this case, the values of normal components of the wave vec-
tors in vacuum,
k
z1
= k
0
2
- k
x
2
1/2
ik
x
, 1a
and in the negative-epsilon medium,
k
z2
=
2
2
k
0
2
- k
x
2
1/2
ik
x
, 1b
as well as the corresponding impedance values for the TM
wave,
1
=
k
z
k
0
ik
x
k
0
,
2
=
k
z
2
k
0
ik
x
2
k
0
, 2
no longer depend on the permeability. Therefore, the sign of
is immaterial and the TM space harmonics are treated by
this lens in the same way as by a slab of left-handed material.
This note does not concern the TE-polarized waves, for
which = k
0
/ k
z
-ik
0
/ k
x
. Further, we will consider the
case of the TM polarization.
To study the lens, we decompose the electromagnetic field
into plane waves and employ the transfer function method.
4,7
For example, for the TM polarization this method implies
that the source and image fields can be presented as
H
y0
xe
-it
=
hk
x
expik
x
x - tdk
x
and
Hx =
hk
x
Tk
x
expik
x
xdk
x
,
respectively, with Tk
x
being the transfer function. To find
the transfer function, it is convenient to resort to the transfer
matrix formalism.
8
The transfer matrix M relates the com-
plex amplitudes of the incident and reflected waves to the
ones on the opposite side of the one-dimensional 1D sys-
tem. For a given polarization, the 2 2 M matrix has ele-
ments m
ij
. One can see that the transfer function is equal to
1/ m
22
. The single-layer-lens transfer matrix, Ml
1
, d , l
2
, cor-
responding to the pass from the source placed at z =0 to the
image plane at z = l
1
+ d + l
2
, is equal to the product
J
1
l
1
SJ
2
dS
-1
J
1
l
2
, where
PHYSICAL REVIEW B 73, 235126 2006
1098-0121/2006/7323/2351266 ©2006 The American Physical Society 235126-1