Photonic Band Gap from a Stack of Positive and Negative Index Materials

Layered heterostructures combining ordinary and negative refractive index materials are shown to display a new type of photonic band gap corresponding to zero (volume) averaged refractive index. Distinct from band gaps induced by Bragg scattering, the zero-$\overline{n}$ gap is invariant upon a change of scale length and is insensitive to disorder that is symmetric in the random variable. A metallic structure that exhibits such a band gap is explicitly designed, and its properties are calculated with accurate finite difference time domain simulations.

Figure 1

(a) Dispersion relationship of a photonic crystal with unit cell consisting of one layer of air ($\mathrm{\text{thickness}}=16\mathrm{m}\mathrm{m}$) and one layer of negative-$n$ material ($\epsilon =-8,\mu =-2$, $\mathrm{\text{thickness}}=4\mathrm{m}\mathrm{m}$); (b) Transmittance through a 1D photonic crystal slab consisting of 32 such unit cells.

Figure 2

(a) Effective $\epsilon$ and $\mu$ of the negative-$n$ material, as given by Eq. (13); (b) Dispersion relationship of a photonic crystal with alternate layers of air (12 mm thick) the negative-$n$ material (6.0 mm thick) with material parameter as shown in (a); (c) Solid line: Transmittance through 16 unit cells, corresponding to the band structure in (b). Dotted line: Transmittance when the lattice constant is scaled by $2/3$; (d) Transmittance through 16 unit cells, with various degree of disorder in thickness. See text for details.

Figure 3

Structural details of the negative-$n$ material.

Figure 4

(a) ${\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ as functions of frequency [12] of the negative-$n$ material shown in Fig. 3; (b) band structure for a photonic crystal with alternating layers of air (7 mm thick) and the designed negative-$n$ material [$\mathrm{\text{thickness}}=3.5\mathrm{m}\mathrm{m}$, ${\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}},{\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ shown in (a)]; (c) Transmittance through a slab consisting of 16 unit cells with details described above, through direct FDTD simulation (open circles) and material properties represented by ${\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and ${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}$ (solid line) [12].