14 Label-Free Super-Resolution Imaging with Hyperbolic Materials
349
These difficulties with the limited signal-to-noise ratio in metamaterial imaging
systems based on direct imaging, can be addressed in the structured illumination
setting, using hyperbolic medium as the substrate [29]. This “hyper-structured illumination” approach employs planar geometry and does not require a thick substrate.
As a result, the effect of loss can be offset by raising the illumination intensity, without
increasing the optical power density in the sample, since the object in a structured
illumination set up is placed “after” the metamaterial, not “before.” Furthermore,
as the hyperbolic medium can support illumination patterns having subwavelength
features, the resulting resolution is also dramatically improved in comparison to the
conventional structural illumination.
In the next sections, we review the relevant properties of the hyperbolic media,
and describe the actual implementation of the hyperlens and the hyperstructured
illumination in practical imaging systems.
14.4 Hyperbolic Media
An isotropic dielectric material is described by a positive (real part of the) permittivity
. Its iso-frequency surface (the constant frequency surface in the wavevector space
k ≡ (k x , k y , k z )), with the corresponding circular iso-frequency curve in (k τ ≡ (k
2
x +
k
2
y )
1/2 sign [k x ] , k n ≡ k z ) coordinates,
k
2
=
ω
2
c 2 ,
(14.1)
where ω is the frequency and c is the speed of light, characterizes the available
states for light propagation within the medium. For a uniaxial anisotropic dielectric
medium with τ = n (where ˆ
n represents the direction of the remaining symmetry
axis), the iso-frequency curve is circular for the ordinary (E ⊥ ˆ
n), or s, and elliptic
for the extraordinary ( ˆ
n · E = 0), or p, polarizations:
k
2
τ + k
2
n
τ
=
ω
2
c 2 , s-polarization,
(14.2)
k
2
τ
n
+
k
2
n
τ
=
ω
2
c 2 , p-polarization.
(14.3)
As a result, in the the dielectric there is a bound on the maximum wavenumber that
can be supported in the material, which leads to diffraction-limited light propagation.
In contrast to this behavior, a metal, with its negative (real part of the) permittivity,
does not support any propagating waves, and the corresponding phase space is empty.
The situation, however, is dramatically different in the uniaxial medium that is so
anisotropic that it shows metallic (highly conducting) behavior in one direction and
dielectric (low conductivity) in the other (such as graphite)—with the corresponding
349
These difficulties with the limited signal-to-noise ratio in metamaterial imaging
systems based on direct imaging, can be addressed in the structured illumination
setting, using hyperbolic medium as the substrate [29]. This “hyper-structured illumination” approach employs planar geometry and does not require a thick substrate.
As a result, the effect of loss can be offset by raising the illumination intensity, without
increasing the optical power density in the sample, since the object in a structured
illumination set up is placed “after” the metamaterial, not “before.” Furthermore,
as the hyperbolic medium can support illumination patterns having subwavelength
features, the resulting resolution is also dramatically improved in comparison to the
conventional structural illumination.
In the next sections, we review the relevant properties of the hyperbolic media,
and describe the actual implementation of the hyperlens and the hyperstructured
illumination in practical imaging systems.
14.4 Hyperbolic Media
An isotropic dielectric material is described by a positive (real part of the) permittivity
. Its iso-frequency surface (the constant frequency surface in the wavevector space
k ≡ (k x , k y , k z )), with the corresponding circular iso-frequency curve in (k τ ≡ (k
2
x +
k
2
y )
1/2 sign [k x ] , k n ≡ k z ) coordinates,
k
2
=
ω
2
c 2 ,
(14.1)
where ω is the frequency and c is the speed of light, characterizes the available
states for light propagation within the medium. For a uniaxial anisotropic dielectric
medium with τ = n (where ˆ
n represents the direction of the remaining symmetry
axis), the iso-frequency curve is circular for the ordinary (E ⊥ ˆ
n), or s, and elliptic
for the extraordinary ( ˆ
n · E = 0), or p, polarizations:
k
2
τ + k
2
n
τ
=
ω
2
c 2 , s-polarization,
(14.2)
k
2
τ
n
+
k
2
n
τ
=
ω
2
c 2 , p-polarization.
(14.3)
As a result, in the the dielectric there is a bound on the maximum wavenumber that
can be supported in the material, which leads to diffraction-limited light propagation.
In contrast to this behavior, a metal, with its negative (real part of the) permittivity,
does not support any propagating waves, and the corresponding phase space is empty.
The situation, however, is dramatically different in the uniaxial medium that is so
anisotropic that it shows metallic (highly conducting) behavior in one direction and
dielectric (low conductivity) in the other (such as graphite)—with the corresponding