We develop a simple analytical model for calculating the vibrational contribution to the nonlinear refractive index n2 (Kerr coefficient) of a crystal in terms of known crystalline parameters such as the linear refractive index, the coefficient of thermal expansion, atomic density, and the reduced mass and the natural oscillation frequency of the vibrational modes of the crystal lattice. We show that the value of this contribution in the terahertz spectral region can exceed the value of the nonlinear refractive index n2 in the visible and near-IR spectral ranges (which is largely electronic in origin) by several orders of magnitude. For example, for crystal quartz the value of the Kerr coefficient in the low-frequency limit is n2 = 2.2 × 10 −9 esu or equivalently is 4.4 × 10 −16 m 2 /W, which is very much larger than its value of 3 × 10 −20 m 2 /W in the visible range. Furthermore, we present an analysis of the dispersion of n2 in the terahertz spectral range and show that even larger values of n2 occur for frequencies close to the vibrational resonances.

A formula is derived for calculating the vibrational contribution to the coefficient of the nonlinear refractive index of a crystal in terms of known crystal parameters such as the coefficient of thermal expansion, the linear refractive index, and the concentration, reduced mass and the natural oscillation frequency of ions in the crystal lattice. It is shown that the value of this coefficient in the terahertz spectral region can exceed its value in the visible and near-IR spectral ranges by several orders of magnitude. Moreover, the dispersion of for crystals in the terahertz spectral range is analyzed.