LSRMDLTR

A non-symplectic numerically integrated planar undulator including optional co-propagating laser beam for laser modulation of the electron beam.
Parallel capable? : yes

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	non-adaptive runge-kutta	integration method (runge-kutta, bulirsch-stoer, modified-midpoint, two-pass modified-midpoint, leap-frog, non-adaptive runge-kutta)
FIELD_EXPANSION	$NULL$	STRING	leading terms	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	0.1557150345504	Strength factor for the first and last pole.
POLE_FACTOR2		double	0.380687012288581	Strength factor for the second and second-to-last pole.
POLE_FACTOR3		double	0.802829337348179	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, $w_0 = \sqrt{2}\sigma_x = \sqrt{2}\sigma_y$
LASER_PHASE	$RAD$	double	0.0	laser phase
LASER_X0	$M$	double	0.0	laser horizontal offset at center of wiggler
LASER_Y0	$M$	double	0.0	laser vertical offset at center of wiggler

A non-symplectic numerically integrated planar undulator including optional co-propagating laser beam for laser modulation of the electron beam.

Parameter Name	Units	Type	Default	Description
LASER_Z0	$M$	double	0.0	offset of waist position from center of wiggler
LASER_TILT	$RAD$	double	0.0	laser tilt
LASER_M		long	0	laser horizontal mode number ($<$5)
LASER_N		long	0	laser vertical mode number ($<$5)
SYNCH_RAD		long	0	Include classical synchrotron radiation?
ISR		long	0	Include quantum excitation?
TIME_PROFILE	$NULL$	STRING	NULL	$<$filename$>$=$<$x$>$+$<$y$>$ form specification of input file giving time-dependent modulation of the laser electric and magnetic fields.
TIME_OFFSET	$S$	double	0.0	Time offset of the laser profile.
GROUP		string	NULL	Optionally used to assign an element to a group, with a user-defined name. Group names will appear in the parameter output file in the column ElementGroup

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)
  

  $$B_x = 0,$$ (22)

  
  
  
  

  $$B_y = B_0 \cosh k_u y \cos k_u z,$$ (23)

  
  
  and
  

  $$B_z = B_0 \sinh k_u y \cos k_u z ,$$ (24)

  
  
  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 $y$:
  

  $$B_y = B_0 ( 1 + \frac{1}{2}(k_u y)^2 ) \cos k_u z,$$ (25)

  
  
  and
  

  $$B_z = B_0 k_u y \cos k_u z.$$ (26)

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

• The ``ideal'' field:
  

  $$B_y = B_0 \cos k_u z,$$ (27)

  
  
  
  

  $$B_z = B_0 k_u y \cos k_u z.$$ (28)

  
  
  This is about 10% faster than the leading-order mode, but less precise. Also, it does not include vertical focusing, so it is not generally recommended.

The expressions for the laser field used by this element are from A. Chao's article ``Laser Acceleration -- Focussed Laser,'' available on-line at
http://www.slac.stanford.edu/$\sim$achao/LaserAccelerationFocussed.pdf . The implementation covers laser modes TEMij, where $0\leq i \leq 4$ and $0 \leq j \leq 4$.

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

$$\lambda_l = \frac{\lambda_u}{2 \gamma^2} \left( 1 + \frac{1}{2} K^2 \right),$$ (29)

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 three pole factors are defined so that the trajectory is centered about $x=0$ and $x^\prime=0$ with zero dispersion. This wouldn't be true with the standard two-pole termination, which might cause problems overlapping the laser with the electron beam.

The laser time profile can be specified using the TIME_PROFILE parameter to specify the name of an SDDS file containing the profile. If given, the electric and magnetic fields of the laser are multiplied by the profile $P(t)$. Hence, the laser intensity is multiplied by $P^2(t)$. By default $t=0$ in the profile is lined up with $\langle t \rangle$ in the electron bunch. This can be changed with the TIME_OFFSET parameter. A positive value of TIME_OFFSET moves the laser profile forward in time (toward the head of the bunch).

Explanation of $<$filename$>$=$<$x$>$+$<$y$>$ format: Several elements in elegant make use of data from external files to provide input waveforms. The external files are SDDS files, which may have many columns. In order to provide a convenient way to specify both the filename and the columns to use, we frequently employ $<$filename$>$=$<$x$>$+$<$y$>$ format for the parameter value. For example, if the parameter value is waveform.sdds=t+A, then it means that columns t and A will be taken from file waveform.sdds. The first column is always the independent variable (e.g., time, position, or frequency), while the second column is the dependent quantity.