Electromagnetic homogenization approximation calculates an effective refractive index of a composite material as a weighted average of its components, and has found uses in gradient refractive index and transformation optics devices. However, the utility of the homogenization approximation is hindered by uncertainty in its range of applicability. Harnessing the capability of time-resolved imaging provided by the terahertz polaritonics platform, we determined the dispersion curves of slab waveguides with periodic arrays of holes, and we quantified the breakdown of the homogenization approximation as the period approached the terahertz wavelength and the structure approached the photonic bandgap regime. We found that if the propagation wavelength in the dielectric waveguide was at least two times as large as the Bragg condition wavelength, the homogenization approximation held independent of the detailed geometry, propagation direction, or fill fraction. This value is much less demanding than the estimate of 10:1 often assumed for homogenization. We further used the experimental capabilities to extract the effective refractive index of the photonic crystals in the homogenization approximation limit, and we used this to analyze the predictive strength of analytical formulas. These formulas enabled rapid design of a Luneburg lens and a bi-directional cloak in a waveguide platform without the need for numerical simulations. Movies of terahertz waves interacting with these structures, which were fabricated using femtosecond laser machining, reveal excellent performance. The combination of an analytical formula and confidence in the homogenization approximation will aid in fast design and prototyping of gradient index devices.
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