$\mathrm{X}$-ray third-order nonlinear Renninger effect and rocking curves

In the work third-order nonlinear Takagi’s equations for $\mathrm{X}$-ray monochromatic waves are investigated for the forbidden reflection case. The forbidden dynamical diffraction in the nonlinear case is related to the presence in the nonlinear equations of the terms proportional to the zero order and the second order nonzero Fourier coefficients of the third-order nonlinear susceptibility. As a consequence, in the third-order nonlinear Bragg diffraction case, a nonlinear analogue of the well known Renninger effect takes place, which is considered theoretically and numerically. The numerical calculations show that in the Bragg geometry the nonlinear reflection curve’s behavior is the same as for not forbidden reflection, but for forbidden reflection the rocking curves are by several orders more sensitive to the input intensity value.