330
I. I. Smolyaninov and V. N. Smolyaninova
structures that could be realized experimentally. Unfortunately, it appears difficult
to develop metamaterials with low-loss, broadband performance. The difficulties are
especially severe in the visible frequency range where good magnetic performance is
limited. On the other hand, recently we have demonstrated that many transformation
optics and metamaterial-based devices, such as electromagnetic cloaks, requiring
anisotropic dielectric permittivity and magnetic permeability could be emulated by
specially designed tapered waveguides [19]. This approach leads to low-loss, broadband performance in the visible frequency range, which is difficult to achieve by other
means. It appears that this approach may be also applied to experimental realization
of the Maxwell fisheye and inverted Eaton microlenses [18], which were suggested
to act as superb imaging devices even in the absence of negative refraction [16].
Realization of these microlenses using electromagnetic metamaterials would require
sophisticated nanofabrication techniques. In contrast, our approach leads to a much
simpler design, which involves two-dimensional (2D) imaging using a small liquid
microdroplet.
Despite strong experimental and theoretical evidence supporting superresolution
imaging based on microlenses and microdroplets, imaging mechanisms involved are
not well understood. Magnification of near-field image components has been suggested in recent experiments with self-assembled plano-spherical nanolenses [24,
25] and high-index liquid-immersed microspheres [26] which demonstrated resolution of the order of λ/4 to λ/7. Our analysis in terms of the effective metamaterial
parameters indicates that the shape of microlenses and microdroplets provides natural realization of the effective refractive index distribution in the fisheye and inverted
Eaton microlenses.
The starting point of our analyses is the dispersion law of guided modes in a tapered
waveguide. In case of the metal-coated dielectric waveguide it can be written in a
simple analytical form:
ω
2 n
2
d
c 2 = k
2
x + k
2
y +
π
2 l
2
d (r) 2
(13.6)
where n d is the refractive index of the dielectric, d(r) is the waveguide thickness, and
l is the transverse mode number. We assume that the thickness d of the waveguide
in the z-direction changes adiabatically with radius r. A photon launched into the lth
mode of the waveguide stays in this mode as long as d changes adiabatically [27].
If we wish to emulate refractive index distribution n(r) of either 2D fisheye or 2D
inverted Eaton lens:
ω
2 n
2
(r)
c 2
= k
2
x + k
2
y
(13.7)
we need to produce the following profile of the microdroplet:
d =
lλ
2
n
2
d − n 2 (r)
(13.8)
I. I. Smolyaninov and V. N. Smolyaninova
structures that could be realized experimentally. Unfortunately, it appears difficult
to develop metamaterials with low-loss, broadband performance. The difficulties are
especially severe in the visible frequency range where good magnetic performance is
limited. On the other hand, recently we have demonstrated that many transformation
optics and metamaterial-based devices, such as electromagnetic cloaks, requiring
anisotropic dielectric permittivity and magnetic permeability could be emulated by
specially designed tapered waveguides [19]. This approach leads to low-loss, broadband performance in the visible frequency range, which is difficult to achieve by other
means. It appears that this approach may be also applied to experimental realization
of the Maxwell fisheye and inverted Eaton microlenses [18], which were suggested
to act as superb imaging devices even in the absence of negative refraction [16].
Realization of these microlenses using electromagnetic metamaterials would require
sophisticated nanofabrication techniques. In contrast, our approach leads to a much
simpler design, which involves two-dimensional (2D) imaging using a small liquid
microdroplet.
Despite strong experimental and theoretical evidence supporting superresolution
imaging based on microlenses and microdroplets, imaging mechanisms involved are
not well understood. Magnification of near-field image components has been suggested in recent experiments with self-assembled plano-spherical nanolenses [24,
25] and high-index liquid-immersed microspheres [26] which demonstrated resolution of the order of λ/4 to λ/7. Our analysis in terms of the effective metamaterial
parameters indicates that the shape of microlenses and microdroplets provides natural realization of the effective refractive index distribution in the fisheye and inverted
Eaton microlenses.
The starting point of our analyses is the dispersion law of guided modes in a tapered
waveguide. In case of the metal-coated dielectric waveguide it can be written in a
simple analytical form:
ω
2 n
2
d
c 2 = k
2
x + k
2
y +
π
2 l
2
d (r) 2
(13.6)
where n d is the refractive index of the dielectric, d(r) is the waveguide thickness, and
l is the transverse mode number. We assume that the thickness d of the waveguide
in the z-direction changes adiabatically with radius r. A photon launched into the lth
mode of the waveguide stays in this mode as long as d changes adiabatically [27].
If we wish to emulate refractive index distribution n(r) of either 2D fisheye or 2D
inverted Eaton lens:
ω
2 n
2
(r)
c 2
= k
2
x + k
2
y
(13.7)
we need to produce the following profile of the microdroplet:
d =
lλ
2
n
2
d − n 2 (r)
(13.8)