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A nonlinear electromagnetic scattering problem is studied in the presence of bound states in the radiation continuum. It is shown that the solution is not analytic in the nonlinear susceptibility and the conventional perturbation theory fails. A non-perturbative approach is proposed and applied to the system of two parallel periodic arrays of dielectric cylinders with a second order nonlinear susceptibility. This scattering system is known to have bound states in the radiation continuum. In particular, it is demonstrated that, for a wide range of values of the nonlinear susceptibility, the conversion rate of the incident fundamental harmonic into the second one can be as high as 40% when the distance between the arrays is as low as a half of the incident radiation wavelength. The effect is solely attributed to the presence of bound states in the radiation continuum.

The resonant nonlinear scattering theory with bound states in the radiation continuum and the second harmonic generation