We rigorously analyze the dispersion function and the curvature of the dispersion surface of a photonic crystal to explore the fundamental limit of its angular sensitivities. With insight gained from group theory, we find that symmetry induced degeneracy gives rise to a singular dispersion surface curvature and a nonvanishing group velocity simultaneously. Near such a singularity, high angular sensitivities can be achieved at low optical loss. This phenomenon exists generally in most common two-dimensional and three-dimensional photonic crystal lattices, although it occurs only for certain photonic bands as dictated by symmetry. This symmetry-induced effect is absent in one-dimensional crystals. Rigorous formulas of the sensitivities of the light beam directions to wavelength and refractive index changes are derived. Individual contributions of the dispersion surface curvature and group velocity to these sensitivities are separated. In the absence of the Van Hove singularity, a singular dispersion surface curvature gives rise to ultrahigh dispersion $\mid \frac{d\theta }{d\lambda }\mid >{10}^3\mathrm{deg}∕\mathrm{nm}$ and refractive index sensitivity $\mid \frac{d\theta }{d{n}_a}\mid >{10}^4\mathrm{deg}$ without compromising optical transmission. The angular dispersion value is significantly larger than those previously reported for the superprism effect and is not due to slow group velocity. We also discuss how various parameters intrinsic and extrinsic to a photonic crystal may suppress or enhance the angular sensitivities according to the rigorous formulas we obtain.

• Received 11 October 2007

DOI:https://doi.org/10.1103/PhysRevB.77.075104