All-Dielectric Rod-Type Metamaterials at Optical Frequencies

Light propagation in all-dielectric rod-type metamaterials is studied theoretically. The electric and magnetic dipole moments of the rods are derived analytically in the long-wavelength limit. The effective permittivity and permeability of a square lattice of rods are calculated by homogenizing the corresponding array of dipoles. The role of dipole resonances in the optical properties of the rod array is interpreted. This structure is found to exhibit a true left-handed behavior, confirming previous experiments [L. Peng et al., Phys. Rev. Lett. 98, 157403 (2007)]. A scaling analysis shows that this effect holds at optical frequencies and can be obtained by using rods made, for example, of silicon.

Figure 1

$|b_0|$ (thick blue solid line), $|b_1|$ (thin gray solid line), and $|b_2|$ (thin red dashed line) for rods with $\epsilon =600$. The insets show the amplitude of the scattered electric field at the first maxima of $|b_0|$ and $|b_1|$.

Figure 2

(Left) Real parts of ${\epsilon }_{zz}$ (thick blue line) and ${\mu }_{yy}$ (thin gray line) of the rod array ($R=0.68a/3$, $\epsilon =600$). (Right) Dispersion curves of the structure calculated from the effective material parameters (black solid lines) and by the PWE method (gray dashed lines).

Figure 3

(Left) Wavelength positions of the two lower-frequency maxima of $|b_0|$ (thick blue lines) and $|b_1|$ (thin gray lines). (Right) $|b_0|$ (thick blue solid line), $|b_1|$ (thin gray solid line) and $|b_2|$ (thin red dashed line) for rods with $\epsilon =12$.

Figure 4

Photonic band structures (a),(c) and second-band IFCs (b),(d) of the higher and lower-$\epsilon$ rod arrays (top: $\epsilon =600$, bottom: $\epsilon =12$). The IFCs are labeled by their reduced frequency $a/\lambda$. (e) Steady-state amplitude of the electric field of light at $\lambda =1.55\mu \mathrm{m}$ incident at 20° on the lower-$\epsilon$ structure ($R=158\mathrm{nm}$, $a=698\mathrm{nm}$).