Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies

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, the 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, 4.4×10^-160.28em0exm²/W, which is very much larger than its value of 3×10^-200.28em0exm²/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 at frequencies close to the vibrational resonances.