Subwavelength imaging with quantum metamaterials

We study the potential of a novel “quantum metamaterial” for subwavelength imaging applications in the midinfrared. Because the layers that comprise the metamaterial have in-plane and out-of-plane dielectric responses that are determined by different physical mechanisms (Drude free electron response and quantized electronic transitions, respectively), their resonances are polarization sensitive and can be designed independently. The result is a negatively refracting anisotropic effective medium with losses, described by the figure of merit, FOM $=$ Re($k_{\perp }$)/Im($k_{\perp }$) ∼ 200 ($k_{\perp }$ is the wave vector), that are significantly lower than metamaterials based on classical layered systems. We find that, with sample design parameters that are realistically achievable with conventional epitaxy technologies, it is possible to obtain negative refraction for all incident angles, and finite element modeling studies indicate that these structures can function as so-called “hyperlenses,” offering low-loss ∼λ/13 spatial resolution at mid-IR wavelengths of λ ∼ 10 μm.

Figure 1

(a) Geometry of the quantum metamaterial and orientation of the polarized electric fields. (b) Schematic of the electronic energy level structure in each QW layer, showing the “intersubband transition” between the confined electron states that generates the tunable peak in the dielectric response. (c) Real and imaginary part of the effective permittivity ${\varepsilon }_{\mathrm{QW},\parallel }$ and ${\varepsilon }_{\mathrm{QW},\perp }$ of a quantum well with 8 nm GaAs thickness. (d) Real part of ${\varepsilon }_{\mathrm{QW},\perp }$ dependence on the GaAs layer thickness.

Figure 2

(a) Calculation of the absorption coefficient α and FOM for a TM incident wave, as a function of the wavelength for the quantum metamaterial of Fig. 1a. (b) Near-field ($\left|\mathrm{Hy}\right|$) calculation of a monochromatic TM Gaussian beam across theair/metamaterial interface, λ $=$ 10.46 μm. The arrows show the direction of the Poynting vector of the transmitted beam, $S_t$, which indicates the negative refraction in the metamaterial, and the associated wave vector, $k_t$.

Figure 3

(a) Isofrequency dispersion curves comparing the dispersion in air and in the $d_{\mathrm{QW}}$ $=$ 8 nm quantum metamaterial slab of Fig. 1, at the wavelength λ $=$ 10.49 μm, where the refraction modeling in Fig. 2 took place. $k_i$ and $k_t$ represent the incident and transmitted wave vectors, respectively, and $S_i$ and $S_t$ are the respective energy flows for incident beam angles of 30° (solid lines) and 60° (dashed lines). (b) Isofrequency dispersion curves for the quantum metamaterial slab showing the transition from near-isotropic to hyperbolic to uniaxial dispersion characteristics as the wavelength is tuned through the negative response region corresponding to the electronic ISBT transitions in the quantum wells. The arrow denotes the propagation direction of the dominant $k_{\parallel }$/$k_0$ ∼ 13.8 spatial frequency components in the image. These propagate as “nondiffracting rays,” almost normal to the metamaterial layers.

Figure 4

(a) Logarithmic plot of the TM beam reflectivity of the $d_{\mathrm{QW}}$ $=$ 8 nm quantum metamaterial slab as a function of wavelength and incident angle, $\Theta$, to the sample normal. Note the pronounced discontinuity in the ∼10.1 μm < λ < ∼11.4 μm spectral region where the negative refraction occurs. (b) Contour plot of the in-plane component of the energy flow inside the quantum metamaterial slab, as a fraction of the in-plane component of the energy flow in the incident beam. Note the negative region appearing at all incident angles in the (∼10.1 μm < λ < ∼11.4 μm) spectral region where ${\varepsilon }_{\mathrm{QW},\perp }$ for the QWs is negative.

Figure 5

Left: (a) $\left|\mathrm{Hy}\right|$ near-field image at the air/metamaterial interface at λ $=$ 9 μm, (b) at 10.9 μm, and (c) at 10.9 μm for a dielectric substrate with $n$ $=$ 1.5. Right: intensity plots along the dashed lines. Note the super-resolved image at (b) when the wavelength is tuned to the spectral region where the negative refraction occurs. The inset shows an illustration of the imaging system, made with a 300-nm-thick gold film with two slits of 200 nm separated (center to center) by 800 nm. The film is deposited on a 0.4-μm-thick quantum-metamaterial substrate composed of $d_{\mathrm{QW}}$ $=$ 8 nm quantum wells and $d_{\mathrm{B}}$ $=$ 8 nm barrier slabs as in Fig. 2.