This notebook was created by Sergey Tomin (sergey.tomin@desy.de). Source and license info is on GitHub. July 2019.

Tutorial N9. Simple accelerator based THz source.¶

In this tutorial we will focus on another feature of the SR module (see PFS tutorial N1. Synchrotron radiation module. Web version), namely the calculation of coherent radiation. Details and limitation of the SR module in that mode can be found in G. Geloni, T. Tanikawa and S. Tomin, Dynamical effects on superradiant THz emission from an undulator. J. Synchrotron Rad. (2019). 26, 737-749

As a first step we consider a simple accelerator with the electron beam formation system (bunch compressor). Undulator parameters are chosen to generate radiation in THz range.

Contents¶

1. Accelerator
   • Lattice
   • Simple compression scenario
   • Tracking up to undulator
2. Coherent radiation from the beam

Accelerator¶

Accelerator includes an accelerator module and linearizer (third harmonic cavity) and a bunch compressor. IN other words we reproduce simplified version of the XFEL injector without the injector dogleg.

Lattice¶

Also we can found the main parameters of the chicane with chicane_RTU(yoke_len, dip_dist, r, type)

Simple compression scenario¶

We consider a simple compression scheme with an accelerator module and third harmonic linearizer and a magnetic chicane. To have full picture of the compression techniques, I would recommend:

• I. Zagorodnov and M. Dohlus, Semianalytical modeling of multistage bunch compression with collective effects
• and [M. Dohlus, T. Limberg, and P. Emma, ICFA Beam

Dynamics Newsletter 38, 15 (2005)](http://www-bd.fnal.gov/icfabd/Newsletter38.pdf)

To compress a bunch longitudinally, the time of flight through some section must be shorter for the tail of the bunch than it is for the head. The usual technique starts out by introducing a correlation between the longitudinal position of the particles in the bunch and their energy using a radio frequency (RF) accelerating system. At the end of a linac which induces an energy chirp $\delta' = \frac{1}{E_0}\frac{dE}{ds}$, the mapping of longitudinal position and relative energy deviation of an electron is

\begin{equation} \begin{split} s_1 &= s_0 \\ \delta_1 &= \delta' s_0 + \delta_{i} \end{split} \end{equation}

where $\delta_{i} = \frac{\Delta E_i}{E_0}$ is not correlated energy spread along the bunch length.

The transformation of the longitudinal coordinate in compressor BC can be approximated by the expression up to first order:

\begin{equation} \begin{split} s_2 &= s_1 - R_{56}\delta_1 = (1 - \delta' R_{56}) s_0 + R_{56}\delta_{i}\\ \delta_2 &= \delta_1 \end{split} \end{equation}$$\quad$$

Taking an ensemble average over all particles in the bunch and by definition $<s_0 \delta_{i}> = 0$, the second moment of the distribution $\sigma_{s_0} = <s_2^2>^{1/2}$ is:

\begin{equation} \sigma_{s_2} = \sqrt{ (1 - \delta' R_{56})^2 \sigma_{s_0} + R_{56}^2\sigma_{\delta_{i}}^2 } \end{equation}

Compression factor is : \begin{equation} C = \frac{\sigma_{s_0}}{ \sigma_{s_2}} \end{equation}

Suppose, uncorrelated energy spread is small and we chose $\delta' = -10$ and $R_{56} = -0.048$ m as we calculated above, in that case the compression factor after the chicane is $$ C = \frac{1}{1 - \delta' R_{56}} = 1.9 $$

The non-linearities of both the accelerating RF fields and the longitudinal dispersion can distort the longitudinal phase space. A higher harmonic RF system can be used to compensate the non-linearities of the fundamental frequency system and the higher order longitudinal dispersion in the magnetic chicanes. To linearize longitudinal phase space, a working point for RF phases and amplitudes must be found for the fundamental frequency and the $n$-th harmonic system (n = 3 for the European XFEL).

The relation between the normalized RF amplitudes

\begin{equation} \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & -k & 0 & -n k \\ -k^2 & 0 & -(n k )^2 & 0) \\ 0 & k^3 & 0 &(n k )^3 \end{bmatrix} \begin{bmatrix} V_1 \cos(\phi_1)\\ V_1 \sin(\phi_1)\\ V_{13} \cos(\phi_{13}) \\ V_{13} \sin(\phi_{13}) \\ \end{bmatrix} = \frac{1}{e} \begin{bmatrix} E_1 - E_0\\ E_1\delta_2' - E_0 \delta_0'\\ E_1\delta_2'' - E_0 \delta_0''\\ E_1\delta_2''' - E_0 \delta_0'''\\ \end{bmatrix} \end{equation}$$\quad$$

In our case we assume initial beam energy $E_0 = 5$ MeV and $\delta_0' = \delta_0'' = \delta_0''' = 0$. $$\quad$$ And final energy we chose $E_1 = 130$ MeV and $\delta_2' = -10$ with $\delta_2'' = \delta_2''' = 0$ So, our vector on the right side will be

\begin{equation} \begin{bmatrix} E_1 - E_0\\ E_1\delta_2' - E_0 \delta_0'\\ E_1\delta_2'' - E_0 \delta_0''\\ E_1\delta_2''' - E_0 \delta_0'''\\ \end{bmatrix} = \begin{bmatrix} 125\\ -1300 \\ 0\\ 0\\ \end{bmatrix} \end{equation} In our case undulator has high K parameter $$ R_{56} = -\frac{L_u}{\gamma}(1 + K^2/2) \approx -0.028 \quad m $$

So, total compression after undulator will be $$ C = \frac{1}{1 - \delta R_{56}} = 4.1 $$

Now we update cavities parameters in the lattice

Generate electron beam¶

Tracking up to undulator¶

Coherent radiation from the beam¶

Beam after undulator.¶

as you can notice, the beam was compressed in the undulator in approximatly in two times as was calculated in the Simple compression scenario

Electron trajectories¶

In some cases, it is worth checking the trajectory of the particle used to calculate the radiation. For this purpose, a special object BeamTraject is attached to the object screen after radiation calculation:

screen.beam_traj = BeamTraject()

To retrieve trajectory you need to specify number of electron what you are interested, for example:

x = screen.beam_traj.x(n=0)