All-angle negative refraction without negative effective index

Negative refraction of electromagnetic waves in ‘‘lefthanded materials’’ has become of interest recently because it is the foundation for a variety of novel phenomena. In particular, it has been suggested that negative refraction leads to a superlensing effect that can potentially overcome the diffraction limit inherent in conventional lenses. These phenomena have been described in the context of an effectivemedium theory with negative index of refraction, and at the moment only appear possible in the microwave regime. To explore the possibility of negative refraction in the optical regime, one may turn to photonic crystals as interesting alternatives. Recent experimental and theoretical works indicate that negative refraction phenomena in photonic crystals are possible in regimes of negative group velocity and negative effective index above the first band near the Brillouinzone center ~G!. However, lower frequencies in the band structure may be more desirable in high-resolution superlensing, as we discuss later in this paper. Here, we show that negative refraction can also be achieved without employing materials with negative effective index. In particular, our focus is on the lowest photonic band near a Brillouin-zone corner farthest from G. Interestingly, this band has a positive group velocity and a positive refractive index, but a negative photonic ‘‘effective mass.’’ We exhibit a frequency range so that for all incident angles one obtains only a single, negative-refracted beam. Such all-angle negative refraction ~AANR! is essential for superlens applications. Although our analysis is general, we study two dimensional ~2D! photonic crystals for simplicity. We begin with TE modes ~in-plane electric field! and consider a square lattice of air holes in dielectric e512.0 ~e.g., Si at 1.55 mm!, with lattice constant a and hole radius r50.35a . To visualize and analyze diffraction effects, we employ wave-vector diagrams: constant-frequency contours in k space whose gradient vectors give the group velocities of the photonic modes. Our numerical calculations are carried out in a plane-wave basis by preconditioned conjugate-gradient minimization of the block Rayleigh quotient using a freely available software package developed in-house. A root finder is used to solve for the exact wave vectors that lead to a given frequency. The results for frequencies throughout the lowest photonic band are shown in Fig. 1.