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 $n_2$ (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 $n_2$ 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 $n_2=2.2\times {10}^{-9}$ esu or, equivalently, $4.4\times {10}^{-16}{\mathrm{m}}^2$/W, which is very much larger than its value of $3\times {10}^{-20}{\mathrm{m}}^2$/W in the visible range. Furthermore, we present an analysis of the dispersion of $n_2$ in the terahertz spectral range and show that even larger values of $n_2$ occur at frequencies close to the vibrational resonances.

Figure 1

Dispersion of (a) $n_2$ (solid red lines) and (b) ${\alpha }_2$ in the terahertz frequency range for crystalline quartz (solid blue lines).

Figure 2

Dispersion of (a) $n_2$ (solid red lines) and (b) ${\alpha }_2$ in the terahertz frequency range for crystalline quartz (solid blue lines). The resonant feature appearing in these graphs is due to the two-photon resonance.