The Physics of Optical Dispersion Equations

We explore the disparate optical dispersion equations used to interpolate the refractive index from measurements at a limited number of spectral lines. Initial 19th-century formulations were based on elastic and electromagnetic theories. Subsequent developments were primarily empirical fits to optical-glass data and are misleading when applied to non-polar semiconductors. We show that the theoretical results of Cauchy, Sellmeier, etc., either classical or quantum-mechanical, follow from linear-response theory via the Kramers-Kronig relations regardless of the mechanism assumed. Their use is limited by the number of parameters required. In contrast, most empirical relations were chosen for simplicity and compatibility with classical ray-tracing. Their reliance on measurements in the visible blurs the roles of electronic and ionic polarization. Moreover, most empirical equations predict indices of odd parity in photon energy, in conflict with time-reversal invariance. A reformulation of the empirical dispersion equations consistent with linear-response theory corrects the parity error and clarifies their application to polar and non-polar materials.