Nonlocal homogenization model for a periodic array of $ϵ$-negative rods

We propose an effective permittivity model to homogenize an array of long thin $ϵ$-negative rods arranged in a periodic lattice. It is proven that the effect of spatial dispersion in this electromagnetic crystal cannot be neglected, and that the medium supports dispersionless modes that guide the energy along the rod axes. It is suggested that this effect may be used to achieve subwavelength imaging at the infrared and optical domains. The reflection problem is studied in detail for the case in which the rods are parallel to the interfaces. Full wave numerical simulations demonstrate the validity and accuracy of the new model.

Figure 1

Medium formed by long thin ENG rods arranged in a square lattice.

Figure 2

(Color online) Reflection of a plane wave by a semi-infinite rod medium. (a) The axes of the rods are parallel to the interface. (b) The axes of the rods are normal to the interface.

Figure 3

Plot of $k_z$ as a function of normalized frequency $\beta a$, for $R=0.05a$ and $k_{\Vert }=0$. The permittivity $ϵ$ follows a Drude-type model (see the text). The solid line represents the “exact” result, whereas the dashed line represents the results calculated with the proposed model.

Figure 4

Plot of $k_z$ as a function of $k_{x}a$ for $R=0.05a$ and $k_y=0$. Contours $\beta a=0.1,0.3,0.5,0.7,0.9$ for the TM modes. The permittivity $ϵ$ follows a Drude-type model (see the text).

Figure 5

Plot of $k_z$ as a function of $k_{x}a$ for $R=0.05a$ and $k_y=0$. Contours $\beta a=1.0,1.2,1.4,1.6,1.8,2.0$ for the TM modes. The permittivity $ϵ$ follows a Drude-type model (see the text).

Figure 6

(Color online) A plane wave illuminates a slab of the rod medium $(N_L=2)$. The rods are parallel to the plane $x=0$. (a) $k_z=0$. (b) $k_y=0$.

Figure 7

Reflection coefficient as a function of the wave vector component parallel to the interface. The slab consists of $N_L=5$ layers, and the rods have $R=0.05a$, and $ϵ=-30$ at $\beta a=1.0$ The solid line represents the full wave MoM data, and the dashed line represents the data obtained using the analytical model.

Figure 8

Reflection coefficient as a function of $k_{y}a$ for $N_L=1$ layer, $R=0.05a$, and $ϵ=-30$ at $\beta a=1.0$. The solid and dashed lines are defined as in Fig. 7.

Figure 9

Reflection coefficient as a function of the free-space wave number $\beta a$ for $R=0.01a$, and $N_L=5$ layers. The permittivity of the rods follows a Drude-type model (see the text). The solid and dashed lines are defined as in Fig. 7.

Figure 10

Reflection coefficient as a function of $k_{z}a$ for the $N_L=5$ layer, $R=0.05a$, and $ϵ=-30$ at $\beta a=1.0$. The solid and dashed lines are defined as in Fig. 7.