LSRMDLTR

A non-symplectic numerically integrated planar undulator including optional co-propagating laser beam for laser modulation of the electron beam.Tracks through a Taylor series map specified by a file containing coefficients.

Parameter Name	Units	Type	Default	Description
L	$M$	double	0.0	length
BU	$T$	double	0.0	Undulator peak field
PERIODS		long	0	Number of undulator periods.
METHOD	$NULL$	STRING	runge-kutta	integration method (runge-kutta, bulirsch-stoer, modified-midpoint, two-pass modified-midpoint, leap-frog, non-adaptive runge-kutta)
FIELD_EXPANSION	$NULL$	STRING	ideal	ideal, exact, or "leading terms"
ACCURACY	$NULL$	double	0.0	Integration accuracy for adaptive integration. (Not recommended)
N_STEPS		long	0	Number of integration steps for non-adaptive integration.
POLE_FACTOR1		double	0.155717533964439	Strength factor for the first and last pole.
POLE_FACTOR2		double	0.380687615192693	Strength factor for the second and second-to-last pole.
POLE_FACTOR3		double	0.80282999969846	Strength factor for the third and third-to-last pole.
LASER_WAVELENGTH	$M$	double	0.0	Laser wavelength. If zero, the wavelength is calculated from the resonance condition.
LASER_PEAK_POWER	$W$	double	0.0	laser peak power
LASER_W0	$M$	double	1	laser spot size at waist
LASER_PHASE	$RAD$	double	0.0	laser phase

This element simulates a planar undulator, together with an optional co-propagating laser beam that can be used as a beam heater or modulator. The simulation is done by numerical integration of the Lorentz equation. It is not symplectic, and hence this element is not recommended for long-term tracking simulation of undulators in storage rings.

The fields in the undulator can be expressed in one of three ways. The FIELD_EXPANSION parameter is used to control which method is used.

• The exact field, given by (see section 3.1.5 of the Handbook of Accelerator Physics and Engineering)
  

  $$B_x = 0,$$ (1)

  
  
  
  

  $$B_y = B_0 \cosh k_u y \cos k_u z,$$ (2)

  
  
  and
  

  $$B_z = B_0 \sinh k_u y \cos k_u z ,$$ (3)

  
  
  where $k_u = 2\pi/\lambda_u$ and $\lambda_u$ is the undulator period. This is the most precise method, but also the slowest.

• The field expanded to leading order in $y$:
  

  $$B_y = B_0 ( 1 + \frac{1}{2}(k_u y)^2 ) \cos k_u z,$$ (4)

  
  
  and
  

  $$B_z = B_0 k_u y \cos k_u z.$$ (5)

  
  
  In most cases, this gives results that are very close to the exact fields, at a savings of 10% in computation time.

• The ``ideal'' field:
  

  $$B_y = B_0 \cos k_u z,$$ (6)

  
  
  
  

  $$B_z = 0.$$ (7)

  
  
  This is about 10% faster than the leading-order mode, but less precise. Small differences from results with the exact field may be seen. Generally, these are too small to be a concern. As a result, this is the default mode.

By default, if the laser wavelength is not given, it is computed from the resonance condition:

$$\lambda_l = \frac{\lambda_u}{2 \gamma^2} \left( 1 + \frac{1}{2} K^2 \right),$$ (8)

where $\gamma$ is the relativistic factor for the beam and $K$ is the undulator parameter.

The adaptive integrator doesn't work well for this element, probably due to sudden changes in field derivatives in the first and last three poles (a result of the implementation of the undulator terminations). Hence, the default integrator is non-adaptive Runge-Kutta. The integration accuracy is controlled via the N_STEPS parameter. N_STEPS should be about 100 times the number of undulator periods.

The expressions for the laser field used by this element were provided by P. Emma (SLAC).