(a) Map of the calculated reflectivity as a function of n and k, the real and imaginary parts of the complex refractive index , of a uniform dielectric film of 180 nm thickness on sapphire for λ = 11.75 μm. The reflectivity drops to zero for ≅ 3.25+1.5i. The black dashed line marks the trajectory of the complex refractive index of VO₂ with increasing temperature. The VO₂ index passes very close to the minimum reflectivity point in n-k parameter space (black dashed line).