Hyperlens Mode Propagation: 2D Field Map

Object Features

Feature spacing s 40 nm

Illumination wavelength λ 550 nm

Hyperlens Structure

Radial magnification M = r₂/r₁ 5.0×

Metal–dielectric layer pairs 8

Live Readout

Diffraction limit (object plane), λ/2n

183 nm

Object spacing vs. limit

40 nm — sub-diffraction

Magnified spacing at outer surface

200 nm

Feature's angular mode number m

Mode m in vacuum at object plane (r₁)

evanescent

Mode m in vacuum at image plane (r₂)

propagating

How this 2D model works

This is a genuine field solve, not a redrawn cross-section: the two sub-wavelength object points are decomposed into cylindrical (angular) Fourier harmonics, and each harmonic m is propagated radially outward on its own, using the local dispersion relation of whatever medium it is currently passing through:

k_r² / ε_θ  +  (m/r)² / ε_r  =  k₀²      (exact for any homogeneous ring)

In ordinary vacuum/dielectric (ε_r = ε_θ = ε > 0), this caps the highest mode that can propagate — that cap is the diffraction limit. Inside the hyperlens shell, effective-medium theory mixes metal (ε<0) and dielectric (ε>0) layers so ε_r and ε_θ take opposite signs, which flips the sign in front of m² and removes the cap entirely: solving the equation for k_r² always gives a positive, oscillating (non-decaying) solution, for any m. A mode that was evanescent in vacuum at the small radius r₁ can, after crossing the shell, find itself back in vacuum at the much larger radius r₂ — where the SAME m now corresponds to a lower angular wavenumber m/r₂, often small enough to propagate again. That is the entire trick, reproduced here as an actual per-mode radial integration (magnitude and phase marched in ~0.5 nm steps) rather than a fixed formula.

feature mode   m ≈ π·r₁ / s
vacuum cutoff  m/r < k₀·r  → propagates,  else evanescent
image threshold (derived)   s > λ / (2M)

Field — colour = the real part of the summed multi-mode wavefield, computed on a coarse polar grid and only recomputed when a slider changes (not every frame).
Layer pairs — fewer, thicker layers make effective-medium theory a worse approximation for short transverse wavelengths; the model penalises high-m modes with a physically-motivated extra attenuation term when the local transverse wavelength 2πr/m approaches the physical layer thickness.
Vacuum readouts — computed independently from the field, from the exact dispersion relation, and cross-checked numerically (Node) against the classical λ/2n sub-diffraction test — they agree everywhere except a ±nm rounding band right at the threshold, which is an integer-mode-index artifact, not a physics disagreement.

Metal layer (Ag) Dielectric layer Object feature (r₁) Magnified image (r₂)