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

A non-symplectic numerically integrated planar undulator including optional co-propagating laser beam for laser modulation of the electron beam.
Parallel capable? : yes
GPU capable? : no
Back-tracking capable? : no

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

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)

  (71)

  (72)

  and

  (73)

  where k_u = 2π∕λ_u and λ_u is the undulator period. This is the most precise method, but also the slowest.

  Experimental feature: One may also model a transverse gradient undulator (TGU) by setting the TGU_GRADIENT parameter to a non-zero value. In this case, taking a as the normalized gradient, the fields are [54]

  (74)

  (75)

  and

  (76)

  where γ is the central relativistic factor for the beam and C is given by the TGU_COMP_FACTOR parameter. This factor, and the term it multiplies, is present in order to help suppress the trajectory error at the end of the device. It may require adjustment in order to achieve the desired level of correction. In addition, the user may need to adjust the pole-strength factors and include external misalignments and steering magnets in order to supress not only the trajectory error, but also dispersion errors.

• The field expanded to leading order in y:

  (77)

  and

  (78)

  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:

  (79)

  (80)

  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.

If HELICAL is set to a nonzero value, a helical device is modeled by combining the fields of two planar devices, one of which is rotated 90 degrees and displaced one quarter wavelength. Again, the FIELD_EXPANSION parameter is used to control which method is used.

• The exact fields are

  (81)

  (82)

  and

  (83)

• The field expanded to leading order in x and y:

  (84)

  (85)

  and

  (86)

• The “ideal” field is

  (87)

  (88)

  (89)

  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/~achao/LaserAccelerationFocussed.pdf . The implementation covers laser modes TEMij, where 0 ≤ i ≤ 4 and 0 ≤ j ≤ 4.

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

(90)

where γ 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^′ = 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 P²(t). By default t = 0 in the profile is lined up with ⟨t⟩ 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.