Nonlinear Optics

Abstract

Many nonlinear phenomena can be conveniently described phenomenologically by expanding the electric polarization P in powers of the light-wave electric field

$$ P_\alpha \; = \;\chi _{\alpha \beta }^{(1)} \;(\omega )\;E_\beta \;(\omega,\text{k})\;\exp \;(\text{i}\text{k}\; \cdot \;\text{x} - \text{i}\omega t) + \chi _{\alpha \beta \gamma }^{(2)} \;(\omega _1,\;\omega _2 )\;E_\beta \;(\omega _1,\text{k}_1 )E_\gamma (\omega _2,\text{k}_2) \times \exp [\text{i}\;(\text{k}_1 + \text{k}_2)\; \cdot \;\text{x} - \text{i}(\omega _1 + \omega _2)\;t] + \chi _{\alpha \beta \gamma \delta }^{(3)} \;(\omega _1,\;\omega _2 \;,\omega _3)\;E_\beta \;(\omega _1,\text{k}_1 )\;E_\gamma \;(\omega _2,\text{k}_2)\;E_\delta \;(\omega _3,\text{k}_3)\; \times \;\exp \;[\text{i}\;(\text{k}_1 + \text{k}_2 + \text{k}_3)\; \cdot \text{x} - \text{i}\;(\omega _1 + \omega _2 + \omega _3)\;t]. $$

(10.1)