Sub-Diffraction-Limit Imaging System with two Interfacing Hyperbolic Metamaterials

It is demonstrated that a system of two interfacing hyperbolic metamaterials may be used for direct subwavelength imaging in the visible range. The proposed configuration consists of two anisotropic photonic crystals: a multilayered metal-dielectric structure and an adjoined wired structure. The results of analytical calculations and numerical modeling show that the light emitted by a dipole placed on the top of the multilayered structure propagates within a narrow cone along the direction $\mathbf{n}$, where $\mathrm{Re}\epsilon (\omega ,\mathbf{n})=0$. Owing to impedance matching and negative refraction at the interface, a bright image with a maximum transverse size of approximately $50$ nm is focused in the wired structure at the wavelength $\lambda =670$ nm.

Figure 1

Two isofrequency surfaces of an extraordinary wave with hyperbolic dispersion. (a) Hyperboloid of two sheets corresponding to type I HMM. It can be realized as a photonic crystal of metallic rods in dielectric matrix. The blue sphere is an isofrequency surface of an ordinary wave. (b) Hyperboloid of one sheet corresponding to type II HMM. Can be realized as a superlattice of metal and dielectric layers.

Figure 2

Scheme of negative refraction at the interface between two hyperbolic media. The incident plane wave, characterized by wave vector ${\mathbf{k}}_i$ and group velocity ${\mathbf{v}}_{g(i)}$, comes from type II HMM ($z>0$) with ${\epsilon }_{\parallel }=3.9$, ${\epsilon }_{\mathrm{\perp }}=-7$. The red hyperbola shows dispersion in this medium given by Eq. (2), where $k_0=\omega /c$. After negative refraction at the interface $z=0$, the plane wave is transmitted to type I HMM ($z<0$) with ${\epsilon }_{\parallel }=-3.9$, ${\epsilon }_{\mathrm{\perp }}=7$ with dispersion shown by the blue hyperbola, Eq. (1). The transmitted wave has the wave vector ${\mathbf{k}}_t$ and group velocity ${\mathbf{v}}_{g(t)}$.

Figure 3

(a) Mutual orientation of the vectors in a plane wave propagating in a uniaxial crystal with optical axis along the dashed line. Pairs of orthogonal vectors are marked by the same color. The direction of Poynting’s vector $\mathbf{S}$ corresponds to a medium with hyperbolic dispersion, as it follows from the direction of group velocity ${\mathbf{v}}_{g(i)}$ in Fig. 2. Angles $\theta$ and $\phi$ are related, $\tan \phi =\mathrm{Re}\left(\frac{{\epsilon }_{\mathrm{\perp }}}{{\epsilon }_{\parallel }}\right)\tan \theta$ [15]. (b) Angular dependence of the dielectric permittivity in a dissipative hyperbolic medium with ${\epsilon }_{\mathrm{\perp }}^{(1)}=-7+i0.5$, ${\epsilon }_{\parallel }^{(1)}=3.9+i0.5$ and in a dissipationless hyperbolic medium with ${\epsilon }_{\mathrm{\perp }}^{(2)}=-7$, ${\epsilon }_{\parallel }^{(2)}=3.9$.

Figure 4

Near-field subwavelength image of a point dipole (shown by white arrow) with $\lambda =670$ nm obtained by negative refraction at the interface between two HMMs. Color map shows spatial distribution of Poynting’s vector. (a) The HMMs satisfy the condition (9) of minimum reflection: ${\epsilon }_{\parallel }^a=3.9$, ${\epsilon }_{\mathrm{\perp }}^a=-7+0.18i$, ${\epsilon }_{\parallel }^b=-3.9+0.18i$, and ${\epsilon }_{\mathrm{\perp }}^b=7+0.26i$. (b) Less dissipative HMMs with parameters not matching the condition of minimum reflection: ${\epsilon }_{\parallel }^a=4$, ${\epsilon }_{\mathrm{\perp }}^a=-7+0.1i$, ${\epsilon }_{\parallel }^b=-2+0.1i$, and ${\epsilon }_{\mathrm{\perp }}^b=9+0.1i$.

Figure 5

Reflection coefficient versus angle of incidence. The green line is for nondissipative media: ${\epsilon }_{\parallel }^a=3.9$, ${\epsilon }_{\mathrm{\perp }}^a=-7$, ${\epsilon }_{\parallel }^b=-3.9$, and ${\epsilon }_{\mathrm{\perp }}^b=7$. The blue line is for weakly dissipative media: ${\epsilon }_{\parallel }^a=3.9$, ${\epsilon }_{\mathrm{\perp }}^a=-7+0.18i$, ${\epsilon }_{\parallel }^b=-3.9+0.18i$, and ${\epsilon }_{\mathrm{\perp }}^b=7+0.26i$. The yellow line is for moderately dissipative media: ${\epsilon }_{\parallel }^a=3.9+0.5i$, ${\epsilon }_{\mathrm{\perp }}^a=-7+0.7i$, ${\epsilon }_{\parallel }^b=-3.9+0.2i$, and ${\epsilon }_{\mathrm{\perp }}^b=7+0.3i$. An enlarged view of the region near the minimum is shown in the inset.

Figure 6

(a) Proposed scheme of the imaging system with layered and wired structures. The black arrow at the top corner is a dipole radiating at $\lambda =670$ nm; (b) distribution of averaged Poynting vector $|⟨\mathbf{S}⟩|$ in the $x$-$z$ plane where the dipole is displaced (at $y=200$ nm); (c),(d) distribution of $|⟨\mathbf{S}⟩|$ in the $x$-$y$ cross-section at the interface $z=0$ and at the focal plane $z=z_f$, respectively.

Figure 7

Transmission of emitted light versus wavelength.