Generation of the second harmonic of soft x rays by reflection from a concave mirror

A theoretical analysis is made of the process of generation of the second harmonic in the case of a nonlinear interaction between three electromagnetic whispering-gallery modes at an abrupt cylindrical plasma–vacuum interface. Formulas are obtained for the efficiency of conversion to the second harmonic in the case when the electromagnetic radiation frequency is considerably higher than the Langmuir frequency of the plasma electrons. Numerical estimates are obtained for soft x rays in the case of grazing incidence on a concave metal mirror.