**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.