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Our structure consists of a square array of alumina rods in air; the calculated band diagram for this structure is shown in Fig. 1a. To obtain negative refraction, equal-frequency surfaces are needed for the photonic crystal that are both convex and larger than those for air (Fig. 1b). Note that conservation of the surface-parallel wave vector gives the direction of the refracted waves inside the photonic crystal (Fig. 1b).

Figure 1: Comparison of positive and negative refraction.
figure 1

a, Band diagram for our structure for transverse magnetic polarization. Shading denotes the frequency range in which negative refraction occurs. b, Equal-frequency contours in k space of air and of the photonic crystal at 13.7 GHz. θ is the angle of incidence from the air to the crystal; vg is the group velocity inside the photonic crystal. c, Negative refraction. Average intensities were calculated at the second interface with the photonic crystal (green) and at the first interface without the photonic crystal (red), and corresponding power distributions were measured. d, Positive refraction. Average intensities were calculated at the second interface with a slab containing polystyrene pellets (blue) and at the first interface without the slab (red), and the corresponding power distributions were measured. Arrows in c and d indicate the refracted beam's direction of divergence from the incident beam.

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We made transmission measurements to confirm our structure's predicted negative refraction, using the interfaces of the photonic crystal in the Γ–M direction. The electric field was kept parallel to the rods for all measurements and calculations; the horn antenna was orientated so that the incident waves make an angle of 45° with the normal of the Γ–M interface. Our structure exhibits the maximum angular range of negative refraction at an operating frequency of 13.7 gigahertz (GHz). In simulations using the finite-difference time-domain method (FDTD), the incident gaussian beam width was set at 6 cm, which is equal to the width of the horn antenna.

The centre of the outgoing gaussian beam is shifted to the left of the centre of the incident gaussian beam (Fig. 1c), which corresponds to negative refraction7. The negative index of refraction was determined to be −1.94, which is close to the theoretical value of −2.06 calculated by the FDTD method. For comparison, we repeated the measurements and simulations with a slab containing only polystyrene pellets, which has a refractive index of 1.46, and found the refracted beam to be on the right of the incident beam, corresponding to a positive index of 1.52 (Fig. 1d).

The advantage of negative refraction in the valence band is that there is no Bragg reflection; such reflections occur in higher bands of the photonic crystal, and we have a well-defined, single-beam propagation that is negatively refracted. Another advantage of operating in the valence band is that the transmission efficiency at this frequency is 63%, which is almost three orders of magnitude larger than the typical transmission efficiency in a left-handed material2,3. The negative-refraction effect that we describe depends only on the refractive index of the dielectric material and on the geometric factors used in two-dimensional photonic crystals. This effect can therefore also be observed at optical wavelengths, at which it is possible to obtain similar refractive indices by using transparent semiconductors.