Third‐order nonlinear two‐wave dynamical X‐ray diffraction in a crystal is considered. For the Laue symmetrical case of diffraction a new exact solution is obtained. The solution is presented via Jacobi elliptic functions. Two input free parameters are essential: the deviation parameter from the Bragg exact angle and the intensity of the incident wave. It is shown that the behavior of the field inside the crystal is determined by the sign of a certain combination of these parameters. For negative and positive signs of this combination, the wavefield is periodic and the nonlinear Pendellösung effect takes place. For the nonlinear Pendellösung distance the appropriate expressions are obtained. When the above‐mentioned combination is zero, the behavior of the field can be periodic as well as non‐periodic and the solution is presented by elementary functions. In the nonperiodic case, the nonlinear case Pendellösung distance tends to infinity. The wavefield diffracts and propagates in a medium, whose susceptibility is modulated by the amplitudes of the wavefields. The behavior of the wavefield can be described also by an effective deviation from the Bragg exact angle. This deviation is also a function of the wavefields.

中文翻译：

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基于新精确解的三阶非线性动力学X射线衍射研究

考虑晶体中的三阶非线性两波动态X射线衍射。对于劳厄对称衍射的情况，获得了一个新的精确解。该解决方案通过Jacobi椭圆函数表示。两个自由输入参数是必不可少的：与布拉格精确角度的偏差参数和入射波的强度。结果表明，晶体内部场的行为由这些参数的某种组合的符号决定。对于这种组合的负号和正号，波场是周期性的，并且会发生非线性Pendellösung效应。对于非线性Pendellösung距离，可以获得适当的表达式。当上述组合为零时 场的行为可以是周期性的也可以是非周期性的，并且解决方案由基本函数表示。在非周期情况下，Pendellösung非线性情况下的距离趋于无穷大。波场在介质中衍射和传播，其介质的敏感性受波场幅度的调制。波场的行为也可以通过与布拉格精确角的有效偏差来描述。该偏差也是波场的函数。