Analysis of a hyperprism for exciting high-$k$ modes and subdiffraction imaging

We study the effect of resonances on the ability of prisms made of hyperbolic metamaterials in the canalization regime (such as wire array media) to couple evanescent high spatial frequencies (high-$k$ modes) to low spatial frequencies that propagate in the far-field zone. Using simple analytical models, we calculate the far-field propagation from the hyperprism. The resonant nature of the metal wire segments within the prism yields a transmission function identical to that of a grating, but with periodicity proportional to the wavelength, making the hyperprism function like a nondispersive grating. Numerically compensating the effect of resonances allows the hyperprism to be used as a one-dimensional imaging device able to resolve feature sizes below the diffraction limit if the host medium has a low refractive index. Furthermore, the hyperprism enables coupling of propagating plane waves to a range of high-$k$ modes that can be increased by increasing the angle of the prism. We quantify how this tunable, nondispersive excitation of high-$k$ modes opens up possibilities for new experimental approaches for coupling to plasmonic systems and for increased axial resolution in total internal reflection imaging, in particular in the terahertz spectrum.

Figure 1

Schematic of the wire array hyperprism. (a) The principle of subdiffraction imaging and surface plasmon excitation are shown in green and red. (b) Two-dimensional representation of the hyperprism with a two-slit object, where $d, 2w, \theta$, and $\phi$ represent the center-to-center distance between slits, prism width, prism angle, and angle of refraction, respectively.

Figure 2

Mapping of input spatial frequencies $k_x/k_0$ to angle of refraction $\phi$ for (a) $n$ = 1 and (b) $n$ = 1.52. Mapping is evaluated for $w\to \infty$. The blue and black curves represent mapping for the hyperprism and a plain isotropic prism, respectively. (c) Mapping of input spatial frequencies $k_x/k_0$ to angle of refraction $\phi$, for $n$ = 1 with two-, three-, and fourfold magnifications.

Figure 3

Normalized amplitude of the output field as a function of input spatial frequencies and angle of refraction for (a) and (b) the nonresonant and (c) and (d) resonant cases. The mapping is evaluated for $n$ = 1 (left column) and $n$ = 1.52 (right column).

Figure 4

Mapping of input spatial frequencies to the angle of refraction for (a) $n$ = 1 and (b) $n$ = 1.52. The mapping is evaluated for $w \to \infty$ and calculated for different grating orders.

Figure 5

(a) Normalized amplitude of the output field as a function of input spatial frequencies. Reconstructed intensity profile for (b) $R$ = 0 and (c) $R$ = 0.9. The intensity profile is evaluated for $n$ = 1. Dashed lines indicate the size and separation of the slits.

Figure 6

Calculated wavelength-dependent intensity profile (a) and (d) for $R$ = 0, (b) and (e) for $R$ = 0.9, and (c) and (f) after convolution. The intensity is evaluated for $n$ = 1 (top row) and $n$ = 1.52 (bottom row). Horizontal dashed lines indicate the size and separation of the slits.

Figure 7

Effective index of surface plasmon polaritons of a single layer of graphene between air and a refractive index of $n=1.52$ as a function of Fermi energy $E_F$ and frequency.

Figure 8

Intensity $1/e$ penetration depth in air as a function of the angle of incidence for a hyperprism with $n=1.52$ at $f=1$ THz.

Figure 9

Calculated transfer kernel of the hyperprism, normalized at each wavelength using (a) and (c) the full-wave 3D simulation and (b) and (d) the Fabry-Pérot resonator model. In the analytical modeling the reflectivity is $R$ = 0.2.