Two-dimensional imaging in hyperbolic media–the role of field components and ordinary waves

We study full vector imaging of two dimensional source fields through finite slabs of media with extreme anisotropy, such as hyperbolic media. For this, we adapt the exact transfer matrix method for uniaxial media to calculate the two dimensional transfer functions and point spread functions for arbitrary vector fields described in Cartesian coordinates. This is more convenient for imaging simulations than the use of the natural, propagation direction-dependent TE/TM basis, and clarifies which field components contribute to sub-diffraction imaging. We study the effect of ordinary waves on image quality, which previous one-dimensional approaches could not consider. Perfect sub-diffraction imaging can be achieved if longitudinal fields are measured, but in the more common case where field intensities or transverse fields are measured, ordinary waves cause artefacts. These become more prevalent when attempting to image large objects with high resolution. We discuss implications for curved hyperbolic imaging geometries such as hyperlenses.


Effective permittivity and local model
For simplicity, the values of relative permittivity for Zeonex (ZNX) and Indium (In) in the narrow range considered in this work (55-58 GHz) are assumed to be non-dispersive, with ε ZNX = 2.3104 + 0.0061i. 1 To estimate the permittivity of indium ε In we use a Drude model with indium plasma frequency ω In = 2π × 2.95 × 10 15 Hz and damping factor γ In = 2π × 4.04 × 10 13 Hz. This yields a Drude model fit consistent with that of Ref. 2 Note that the conductivity of indium σ In = ωε 0 ε i In at the limit of a constant field yields the measured value for the DC conductivity of indium, Applying this Drude model to the frequencies considered here yields ε In ≈ −5 × 10 3 + 4 × 10 6 i.
In the low terahertz range, it can be shown 3 that this wire medium is very well modeled by an effective permittivity tensor with components (Cf. Eq. (1) of the manuscript) ε t = ε h and where f V = πR 2 /a 2 is the volumetric filling fraction of the cylindrical wires, k 0 = ω/c, ε h = ε ZNX and ε m = ε In are the host and metal permittivities, respectively. The plasma wave vector k p is related to the geometry of the wire array as follows: Furthermore, the wire medium supports two modes with propagation constants given by, 3 where k = k 2 x + k 2 y and Here, k z is mostly real, but k (2) z is mostly imaginary, corresponding to rapidly evanescent waves that do no contribute to imaging. 3 Using the additional boundary conditions for a wire medium, 4 it is then possible to obtain the transmission coefficient T for the TM modes of a wire medium slab suspended in air by solving Eq. (19) in Supplementary Ref. 3. Supplementary Fig. S1 shows the transmission amplitude as a function of k using this method for the wire medium considered here, at the three frequencies of interest.
This model yields a material-, geometry-, and nonlocal [i.e. (k x , k y )-dependent] transverse permittivity z yields a spatially dispersive value of ε z (k x , k y ) which has a minimum magnitude corresponding to ε z = −1 × 10 2 + 5 × 10 4 i. Since it has been shown that imaging performance is largely insensitive to |ε z | when it is orders of magnitude larger than the transverse component, 5 we thus use a local description for the permittivity tensor of the metamaterial slab with ε t = 2.3104 + 0.0061i and ε z = −1 × 10 2 + 5 × 10 4 i. Indeed, the transmission properties of a local metamaterial slab of length 3.4 mm using these parameters are in excellent agreement with the full non-local model, as shown in Supplementary Fig. S1 (dotted lines).

Comparison with finite element method
We compare the fields calculated using the transfer matrix method presented here, with those calculated using a commercially available 3D finite element solver (COMSOL). As an illustrative example, we consider an x-polarized point dipole at a frequency of 300 GHz (λ = 1 mm), placed ad a distance of 250 µm (λ /4) away from an anisotropic slab with electric permittivity components ε t = 2 + 0.1i, ε z = 10 and thickness 0.8 mm. For the 3D finite element calculations we use a slab with dimensions 0.8 mm × 10 mm × 10 mm, and a maximum mesh element of 100 µm due to memory constraints (128 GB RAM). The electric fields at the slab output surface are shown in Supplementary Fig. S2, showing excellent agreement between our method and COMSOL. The transfer matrix method is significantly faster (calculations take minutes instead of hours) and can be performed on any desktop computer.

Curved hyperlens
We now compare the imaging performance of a hyperbolic medium slab with that of a curved magnifying hyperlens. A finite-element simulation of a 3D spherical curved hyperlens is not feasible using the large anisotropic material parameters used throughout this work, due to memory constraints and convergence issues. However, some insight on how our conclusion from planar simulations apply to magnifying hyperlenses can be obtained by perfoming finite-element simulations of a curved geometry in 2D, corresponding to a cylindrical hyperlens. The results are shown in Supplementary Fig. S3. The geometry is composed of a hollow hyperbolic material cylinder of inner diameter 0.2 mm and outer diameter 3.6 mm (corresponding to 3.4 mm radial propagation through the hyperbolic medium and a geometric magnification of 18), radial permittivity ε r = −1×10 2 +5×10 4 i, and azimuthal permittivity ε θ = 2.3104+0.0061i, in an air background. A point source is centred at a distance 50 µm below the inner wall of the hyperlens. The extraordinary wave behaviour is obtained by exciting an electric dipole along x, and plotting the outof-plane magnetic field H y (purely transverse-magnetic waves), where propagation and magnification of high spatial frequencies is expected. Results at 55 and 58 GHz are shown in Supplementary Fig. S3(a) and (b), respectively. Note that the magnifying point spread function for extraordinary waves should only weakly depend on frequency, since high azimuthal spatial frequencies are converted to low spatial frequencies upon magnification: Spatial frequencies up to k 0 /36 on the narrow end can still couple to free space on the wide end, and thus do not lead to strong resonances. As a result, the image scrambling observed in a hyperbolic medium slab at frequencies slightly below Fabry-Perot resonance should be reduced in an equivalent curved structure. Indeed, point source magnification is qualitatively similar at both frequencies; a slightly narrower signal with small additional side-oscillations appears at the output surface of the hyperlens at 55 GHz when compared to 58 GHz [ Supplementary Fig. S3(c) and (d)]. Ordinary wave behaviour is analysed using excitation by a magnetic dipole along x, and plotting the out-of-plane electric field E y , (purely transverse-electric waves, for which high spatial frequencies do not propagate). In this case, a strongly diffracted field is observed at the output [ Supplementary Fig. S3(c) and (d)], suggesting that in a 3D geometry the output signal would be strongly asymmetric. However, to fully comprehend magnifying imaging performance of a curved hyperbolic structure, a complete analysis should be conducted in a spherical basis. This will be the object of future study. Simulations results for 2D cylindrical hyperlens with radial permittivity ε r = −1 × 10 2 + 5 × 10 4 i, and azimuthal permittivity ε θ = 2.3104 + 0.0061i, in an air background. A point dipole source is centred at a distance 50 µm below the inner wall of the hyperlens. The extraordinary field magnitude |H y | is shown for an x-polarized electric dipole at (a) 55 GHz and (b) 58 GHz. Also shown are the field magnitudes along the input (blue) and output surface for extraordinary waves (green) and ordinary waves (red) at (c) 55 and (d) 58 GHz. Curves for ordinary waves correspond to |E y | for an x-polarized magnetic dipole in the same position.