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在经典力学框架内和偶极近似下,引入正弦平方势讨论了系统非线性对摆动场辐射的影响。首先,把粒子在弯晶中的运动方程化为具有周期外力作用的摆方程,用Jacobian椭圆函数和椭圆积分解析地描述了粒子运动行为;用多尺度度法和三角函数的广义Bessel展开,找到了受迫摆方程的近似解。而后,在超相对论情况下讨论了系统的瞬时辐射强度和平均辐射强度,并对两种辐射的强度比进行了讨论。结果表明,摆动场辐射强度与摆动场振幅平方成正比,振幅越大摆动场辐射强度越强。
In the framework of classical mechanics and the dipole approximation, the influence of the system nonlinearity on the radiation of the swing field is discussed. First of all, the motion equation of particles in bending crystal is transformed into a pendulum equation with periodic external force. The particle motion behavior is analytically described by Jacobian elliptic function and elliptic integral. By using the multi-scale method and the generalized Bessel expansion of trigonometric function, The approximate solution to forced pendulum equation. Then, the instantaneous radiation intensity and the average radiation intensity of the system are discussed in the context of the theory of relativity, and the intensity ratio of the two radiation types is discussed. The results show that the radiation intensity of the swing field is proportional to the square of the amplitude of the swing field. The stronger the amplitude, the stronger the radiation intensity of the swing field.