Nonlinear optical phenomena are of particular interest, since they are extremely sensitive to the symmetry of the system, to the features of electronic spectrum, and to the properties of light-matter interaction. The report presents the theory of second harmonic generation on excitons in semiconductors, with the special attention paid to copper oxide crystals, where the exciton effects manifest themselves most vividly. The plan of the report as follows:
1. The symmetry analysis.
2. Two-photon absorption in semiconductors.
3. The mechanism of second harmonic generation in a noncentrosymmetric crystal.
4. Quadrupole and magnetodipole second harmonic in centrosymmetric copper oxide.
5. Polariton effects in the second harmonic. ​

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