Negative Refraction and Left-Handed Electromagnetism in Microwave Photonic Crystals

We demonstrate the negative refraction of microwaves in a metallic photonic crystal prism. The spectral response of the photonic crystal prism, which manifests both positive and negative refraction, is in complete agreement with band-structure calculations and numerical simulations. The validity of Snell’s law with a negative refractive index is confirmed experimentally and theoretically. The negative refraction observed corresponds to left-handed electromagnetism that arises due to the dispersion characteristics of waves in a periodic medium. This mechanism for negative refraction is different from that in metamaterials.

Figure 1

(a) Schematic diagram of microwave refraction experimental setup (not to scale). (b) Propagation vectors for positive and negative refraction. (c)–(f) Microwave electric field maps in the far field region. (c) Negative and (e) positive refraction by the metallic PC prism for the incident beam along $\Gamma \to K$ forming an angle of incidence $30°$ with the refraction surface. $\mathrm{WF}=\mathrm{\text{wave}}\mathrm{\text{front}}$ with respect to refracting surface. (d) Negative refection for the incident beam along $\Gamma \to M$ forming an angle of incidence $60°$. (f) Positive refraction by a polystyrene prism. In all the field maps, approximate area of each field map is $43\times 40{\mathrm{cm}}^2$; the PC prisms and incident beams are schematic and do not correspond to the actual sizes used.

Figure 2

(a) Band structure computed for the triangular lattice metallic PC with $r/a=0.2$. (b) Microwave transmission amplitude $|S_{21}|$ vs frequency $f$ (GHz) on the far side. Note the band cutoff below 6.2 GHz, transmission between 5.8–11.1 GHz, and band gap region 11.1–11.3 GHz are in excellent agreement with Fig. 2a. (c) Refractive index $n_p$ determined from the experimental results for a beam incident along $\Gamma \to K$ ($\Delta$) and $\Gamma \to M$ (*). For $n_p<0$ ($n_p>0$), the electromagnetism is left handed (right handed), and the refraction angle is negative (positive). The close match between the experiment ($\Delta$ and *) and theory (dashed and solid lines) is evident and shows agreement with Snell’s law.

Figure 3

(a) The EFS for the first and second bands. The centers of the hexagons represent the centers of the first Brillouin zones in the respective bands. Blue (red) color represents lower (higher) frequency. (b),(c) Simulations of wave refraction showing the wave front emerging from a metallic PC: (b) positive refraction at 6.6 GHz and (c) negative refraction at 9.7 GHz. The PC used in the simulations has the same size as that used in the experiment. The electric field is plotted as $E_z^{1/3}$ for better visibility. The agreement with Figs. 1e, 1c is evident.