Sub-wavelength imaging by space dilation
Mircea Giloan and Robert Gutt

Company for Applied Informatics, Republicii nr. 107, 400489 Cluj-Napoca, Romania

Abstract: Two transformation-optics inspired flat lenses are used to build up an optical
system capable to transpose an area surrounding the object focal point in a magnified area
surrounding the image focal point. The object and image focal points of the system are placed in the
free space and respectively a dilated space having a sub-unitary relative permittivity and
permeability. The anisotropic and inhomogeneous media constituting the lenses enable the
processing of the high spatial frequencies waves without converting them in evanescent waves.
Numerical simulations show the capability of the proposed device to perform magnified discernible
images of the sub-wavelength details.

Projective imaging systems based on lenses are the best option for high-speed optical
microscopy. The resolution of these optical devices is limited due to the fact that sub-wavelength
information from an object is carried by high spatial frequency waves which become evanescent
inside of a conventional optical lens. Within the last two decades the fields of plasmonics and
metamaterials provide solutions for unconventional lens designs able to process the waves of high
spatial frequencies. This has provided tremendous opportunities for designing optical imaging
devices with unprecedented resolution. Beginning with the seminal concept of the perfect lens [1] a
number of metamaterial based lenses providing sub-wavelength resolution were studied both
theoretically and experimentally [2-11].
   Transformation-optics approach opened a path for designing anisotropic and inhomogeneous
media, capable to manipulate not only the wave paths but also the wave vectors [12-15]. Recently, a
general method was proposed for designing both converging and diverging flat lenses made of
media that perform specific coordinate transformations [16]. The permittivity and permeability
tensors of the transformation media constituting these unconventional lenses are retrieved from the
transformation functions which alter the coordinates of the original space. A converging lens of this
type provides a perfect convergence into its focal point of a light beam parallel to its optical axis.

In this theoretical study, we propose an optical imaging system based on two converging
transformation-optics inspired flat lenses which provides sub-wavelength resolution.
   Considering the z-axis as the optical axis of the proposed system, the transformation media
of the lenses used to build up the device are generated by functions which transform only the z
coordinate of the original space while x and y coordinates remain unaltered. Reference 16 describes
in detail the steps leading to the retrieval of the transformation function which generates the
medium of a converging lens embedded in the free space, i.e. relative permittivity and permeability
are equal to unity. Following the same steps one can easily prove that for a converging lens of
thickness d and focal distance φ embedded in an isotropic and homogeneous medium having the
same relative permittivity and permeability, ε=μ=m, the lens medium is generated by the following
transformation function:

h(x , y)=m[δ−γ (φ2+x2+ y2 )1 /2] , (1)

where δ=1+φ /d and γ=1/d . When m=1 we are in the case of a lens embedded in the free
space (ε=μ=1). The cases when parameter m is supra-unitary (m>1) or sub-unitary (m<1) can be
viewed as an additional transformation of the space like a compression or dilation, respectively.
Regardless the value of parameter m the optical parameters (permittivity and permeability) of the
lens generated by transformation function described by eq. 1 are positive inside the area delimited
by the circle given by equation:

x2+ y2=φ 2ρ(ρ+2) , (2)

where ρ=d /φ is the thickness to focal distance ratio of the lens.
The concept of the optical system presented in this paper is schematically represented in
figure 1. The (y-z)-plane view of two identical converging flat lenses embedded in the free space
(green rectangles) is depicted in figure 1(a). The lenses have a thickness to focal distance equal to
four (ρ=4). Since the proposed device is confined to positive optical parameters of the constituent
lenses the depicted area is limited on y-axis by the singularity point:

ys=φ (ρ(ρ+2))1/2 , (3)

derived from equation 2. The left and right lenses, depicted by red and blue rectangles, correspond
to negative and positive values of z-coordinate and will be named the object and image lenses,
respectively. The geometrical parameters, like focal distance (φ), thickness (d), focal point (F), and
outside plane (P), are denoted for the object and image lenses with the subscripts o and i,
respectively. The curves of constant transformed z coordinate are depicted by red and blue lines
inside the area of the object and image lenses respectively.