Optical anisotropic metamaterials: Negative refraction and focusing

We design three-dimensional (3D) metallic nanowire media with different structures and numerically demonstrate that they can be homogeneous effective indefinite anisotropic media by showing that their dispersion relations are hyperbolic. For a finite slab, a nice fitting procedure is exploited to obtain the dispersion relations from which we retrieve the effective permittivities. The pseudo focusing for the real 3D wire medium agrees very well with the homogeneous medium having the effective permittivity tensor of the wire medium. Studies also show that in the long-wavelength limit, the hyperbolic dispersion relation of the 3D wire medium can be valid even for evanescent modes.

Figure 1

(Color online) (a) Top graph: circular EFS for vacuum and isotropic media. Bottom graph: Equifrequency surfaces for vacuum (circle) and negative anisotropic refraction media (hyperbolic relation). (b) The definitions for $k_{\perp }$, $k_{\parallel }$, ${\epsilon }_{\perp }$, and ${\epsilon }_{\parallel }$ used in our simulations.

Figure 2

(Color online) Material parameters of the metallic layers: $\epsilon =-4$ and $\mu =1$; working frequency $f=0.5\text{GHz}$; space period of metallic layers is 0.06 m; width of metallic layers is 0.02 m. (a) The magnetic field distribution of the group negative refraction in the metallic-air layer array slab. A Gaussian beam with the transverse magnetic polarization and an incident angle of $30°$ is incident on the simulated metallic-air superlattice. The white line indicates the ray-tracing result. (b) A ray-tracing diagram showing that the rays coming from a line source are refocused by an anisotropic medium slab with the effective permittivity $\epsilon$ tensor of our simulated metallic-air layer array slab. (c) The magnetic field distribution of the pseudofocusing of our simulated metallic-plates array slab with a line source placed 1.25 m from the interface, which launches a cylindrical transverse magnetic polarized wave. The thickness and the width of the metallic-air layer array slab are 2.4 m and 6 m, respectively. (d) The magnetic field distribution of the pseudofocusing in a homogeneous anisotropic slab with the effective permittivity of our simulated metallic-air layer array slab $\left({\epsilon }_{\parallel }=1.7293,{\epsilon }_{\perp }=-0.7907\right)$.

Figure 3

(Color online) (a) Schematic of 3D metallic wires embedded in a dielectric matrix. (b) The magnetic field distribution for the negative refraction in a 3D gold-wire square-lattice medium with vacuum background and the wavelength $\lambda =700\text{nm}$. The incident plane wave has transverse magnetic polarization and an incident angle of $45°$. The permittivity $\epsilon$ for gold is taken from experimental data (Ref. 24) $\epsilon =-15.5931+i1.2734$ at $\lambda =700\text{nm}$. The radius, the length of gold wires, and the lattice constant are 16, 1532, and 70 nm, respectively. The white arrow indicates the direction of power flow.

Figure 4

(Color online) The magnetic field distribution of the pseudofocusing in a 3D gold-wire square lattice medium with a line source placed 884 nm away from the interface, which launches a cylindrical transverse magnetic polarized wave at the wavelength $\lambda =700\text{nm}$. The permittivity of gold is the same as in Fig. 3. The background is vacuum. The radius, the length of gold wires, and the lattice constant are 16, 2732, and 70 nm, respectively. The white arrow indicates the direction of power flow.

Figure 5

(Color online) The numerical dispersion-relation data from the simulation (solid circles) and the fitted hyperbolic curve (dashed line). All parameters are the same as in Fig. 4 except the length of gold wires which is 1500 nm. Note that all $k$ components here are normalized by $k_0$, where $k_0=\omega /c$. The inset shows, as a typical example, the field distribution for ${\theta }_i=30°$ fitted by Eq. (2).

Figure 6

(Color online) The effective permittivity ${\epsilon }_{\perp }$ and ${\epsilon }_{\parallel }$ calculated from Maxwell-Garnett equations (solid lines) and numerical simulations (squares) for different wire radii. The simulated medium is a 3D square-lattice silver-wire medium in vacuum with the lattice constant $a=20\text{nm}$. The wavelength is $\lambda =700\text{nm}$. The permittivity of silver at $\lambda =700\text{nm}$ is ${\epsilon }_{\text{silver}}=-20.4373+i1.2863$, taken from experimental data (Ref. 24).

Figure 7

(Color online) (a) The magnetic field distribution of the focusing simulation for the simulated 3D gold-wire square-lattice anisotropic medium slab with the source 884 nm away from the first interface. (b) Same as (a), but for a homogeneous anisotropic slab with the fitted effective parameters ${\epsilon }_{\parallel }=1.4455+i0.0044$ and ${\epsilon }_{\perp }=-1.9082+i0.2391$. (c) and (d) are the same as (a) and (b), respectively, but for the magnetic field intensity distribution. (e)–(h) are the same as (a)–(d), respectively, except the source is 442 nm away from the first interface. All material parameters are the same as in Fig. 4.

Figure 8

(Color online) The magnetic field distribution in a 3D silver-wire hexagonal-lattice medium slab with the alumina background. The incident plane wave has the transverse magnetic polarization and the wavelength in vacuum $\lambda =700\text{nm}$. (a) Normal incidence. (b) At an incident angle of $30°$. The white arrow indicates the direction of power flow. The hexagonal-lattice constant $a$, the radius $r$, and the length $l$ of silver wires are 120, 30, and 1700 nm, respectively. The permittivities of silver and alumina at the wavelength in vacuum $\lambda =700\text{nm}$ are ${\epsilon }_{\text{silver}}=-20.4373+i1.2863$ and ${\epsilon }_{{\text{Al}}_2{\text{O}}_3}=3.1$, respectively, taken from experimental data (Ref. 24).

Figure 9

(Color online) The numerical dispersion (solid circles) and the fitted dispersion curve (dashed line) of 3D silver-wire hexagonal-lattice media in the alumina background. (a) The lattice constant $a=120\text{nm}$ and the radius of silver wires $r=30\text{nm}$. (b) The lattice constant $a=30\text{nm}$ and the radius of silver wires $r=12\text{nm}$. $k_{\parallel }\le k_0$ corresponds to the propagating modes in the background, while $k_{\parallel }>k_0$ corresponds to the evanescent modes. All other parameters are the same as in Fig. 8. Note that all $k$ components are normalized by $k_0$, where $k_0=\sqrt{\epsilon }\omega /c$ and $\epsilon$ is the permittivity of alumina.