Frequency moved undulator

Smilei or EPOCH

Mark the oscillate electrons, and save the field produced by them. Then sparated them from the total electric field and save to a new file.

光波荡器推导

假设平面波激光方向为$\hat k = \frac{\mathbf{k}}{k_L}$，则 $B = \frac{(\hat k \times E)}{c}$，电子在激光场中运动方程写为：

$$
\frac{d}{dt}(\gamma mc \beta) = -e(E + v \times B)
$$

$$
\frac{d}{dt}(\gamma mc \beta) = -e[E + \beta \times (\hat k \times E)] = -e[E + (\beta \cdot E)\hat k - (\beta \cdot \hat k)E]
$$

我们假设激光沿着x方向偏振，$E = E_0 \sin(\omega_Lt − k \cdot x)\hat x$，沿着z方向传播，与z轴呈夹角$\varphi$，波矢$k = k_L(0, -\sin \varphi, +\cos \varphi)$，并且 $\omega_L = ck_L$。运动方程写为：

$$
\frac{d}{dt}(\gamma mc \beta) = -eE_0\sin(ck_Lt - \mathbf{k} \cdot x)[1 - (\hat k \cdot \beta)] = \frac{eE_0}{ck_L}\frac{d}{dt}\cos(ck_Lt - \mathbf{k} \cdot x)
$$

方程表明水平方向动量守恒，通过积分可以很容易得出，

$$
\beta_x = \frac{eE_0}{\gamma mc^2k_L}\cos(ck_Lt - \mathbf{k}\cdot x)
$$

波荡器参数为$K = \frac{eE_0}{mc^2k_L}$，对于Undulator， $K<<1$，$t \approx c/\beta_z$，横向振动速度为$\cos(k_L(1/\overline{\beta}_z - \cos\varphi)z + k_Ly\sin\varphi)$，则波荡器周期为：

$$
\lambda_u \rightarrow \frac{\overline{\beta}_z\lambda_L}{1 - \overline{\beta}_z\cos\varphi}
$$

将波荡器周期和参数K带入波荡器辐射公式,

$$
\frac{\lambda_1(\phi)}{c} = \frac{\lambda_u}{c}[\frac{1 + K^2/(4\gamma^2)}{\beta} - (1 - \frac{\phi^2}{2})] \approx \frac{\lambda_u}{c}\frac{1 + K^2/2 + \gamma^2\phi^2}{2\gamma^2}
$$

同步辐射共振波长为

$$
\lambda = \frac{1 + K^2/2}{2\gamma^2}\frac{\lambda_L}{1 - \overline{\beta}_z\cos\varphi}
$$

当满足$\varphi \rightarrow 0, K << 1$(undulator的假设)时

$$
\lambda \rightarrow \lambda_L
$$

时间收缩效应

假设$t^{‘}$为粒子静止坐标系，则 $1 - \beta(t^{‘})\cos\phi(t^{‘})$ 为时间收缩因子，对于相对论电子，

$$
\beta = \sqrt{1 - \frac{1}{\gamma^2}} \approx 1 - \frac{1}{1\gamma^2}
$$

对于$\gamma >> 1$，$\phi << 1$

$$
1 - \beta \cos\phi \approx \frac{1}{2}(\frac{1}{\gamma^2} + \phi^2)
$$