10.21 CSBEND—A canonical kick sector dipole magnet.

A canonical kick sector dipole magnet.
Parallel capable? : yes
GPU capable? : yes
Back-tracking capable? : yes

A canonical kick sector dipole magnet.

A canonical kick sector dipole magnet.

A canonical kick sector dipole magnet.

A canonical kick sector dipole magnet.

A canonical kick sector dipole magnet.

This element provides a symplectic bending magnet with the exact Hamiltonian. For example, it retains all orders in the momentum offset and curvature. The field expansion is available to eighth order.

One pitfall of symplectic integration is the possibility of orbit and path-length errors for the reference orbit if too few kicks are used. This may be an issue for rings. Hence, one must verify that a sufficient number of kicks are being used by looking at the trajectory closure and length of an on-axis particle by tracking. Using INTEGRATION_ORDER=4 is recommended to reduce the number of required kicks.

As of version 28.0 and later, the REFERENCE_CORRECTION feature is available to compensate for errors inherent in the numerical integration of the trajectories. In particular, depending on the number of kicks used, as well as the bending radius and angle, an on-axis particle may emerge from the element with a non-zero trajectory and a path-length error. With REFERENCE_CORRECTION set to a non-zero value, these errors are subtracted from the coordinates of all particles. There are some pitfalls to using this feature: first, one may not realize that the number of kicks is too small to provide good results, since the output trajectory of the central particle will always be (nearly) identically zero. Second, in a magnet with a gradient or other field nonuniformities, a particle may emerge centered on the ideal trajectory yet still see the impact of the gradient, sextupole, etc. For these reasons, this feature should be used with caution and only when the residual trajectory is large enough to cause problems.

Higher-order field components

Normally, one specifies the higher-order components of the field with the Kn, with n = 1 through 8. The field expansion in the midplane is B_y(x) = B_o * (1 + ∑ _n=1⁸xⁿ). By setting the USE_bN flag to a nonzero value, one may instead specify the bn parameters, which are defined by the expansion B_y(x) = B_o * (1 + ∑ _n=1⁸xⁿ). This is convenient if one is varying the dipole radius but wants to work in terms of constant field quality.

Setting NONLINEAR=0 turns off all the terms above K_1 (or b_1) and also turns off effects due to curvature that would normally result in a gradient producing terms of higher order.

The EXPANSION_ORDER parameter controls the order of the expansion of the nonlinear fields, so that terms are limited to xⁱy^j with i + j ≤EXPANSION_ORDER. By default, when EXPANSION\_ORDER=0, the expansion order is set automatically, as follows: If the highest non-zero multipole order (specified by Kn, Bn, Fn, or Gn) is n (with n = 1 being quadruople), then the expansion order is set to n + 3. However, the expansion order is never automatically set to less than 4, unless all the multipole terms are zero, in which case the expansion always yields a constant. Since the number of polynomial terms increases like the square of the expansion order, using many multipole terms can significantly increase run time. The maximum value for the expansion order is 10.

Edge angles and edge effects

Some confusion may exist about the edge angles, particularly the signs. For a sector magnet, we have of course E1=E2=0. For a symmetric rectangular magnet, E1=E2=ANGLE/2. If ANGLE is negative, then so are E1 and E2. To understand this, imagine a rectangular magnet with positive ANGLE. If the magnet is flipped over, then ANGLE becomes negative, as does the bending radius ρ. Hence, to keep the focal length of the edge 1∕f = -tanE_i∕ρ constant, we must also change the sign of E_i.

Several models are available for edge (or fringe) effects. Which is used depends on the settings of the EDGE_ORDER, EDGE1_EFFECTS, and EDGE2_EFFECTS parameters EDGE1_EFFECTS controls entrance edge effects while EDGE2_EFFECTS controls exit edge effects, as follows:

• 1: — Edge effects using non-symplectic method [3].
  • EDGE_ORDER<2 — linear edge focusing with δ-dependence to all orders. Generally not recommended if symplecticity is important, though when the edge effects are weak it appears acceptable.
  • EDGE_ORDER>=2 — second-order matrix edge focusing with δ-dependence to all orders. Use of this model is strongly discouraged when symplecticity matters.
• 2: — Edge effects using K. Hwang’s method [45]. Note that there will be a trajectory offset when using this method that is particularly evident for small bending radii, due to extension of the fringe field outside the body of the magnet. To suppress this, adjustment of the FSE parameter can be performed automatically if FSE_CORRECTION is set to a non-zero value. If FSE_CORRECTION=1, the path-length is adjusted to match the nominal length, which is not physical; this behavior can be suppressed by setting FSE_CORRECTION=2.
  • EDGE_ORDER<2 — include only terms linear in transverse coordinates, but δ-dependence to all orders. Recommended for applications where symplecticity matters. This method is symplectic.
  • EDGE_ORDER>=2 — include all terms. This settings has been observed to produce emittance damping in some cases (particularly with large emittance and small bending radii), so users are advised to be cautious. This method is non-symplectic.
• 3: — Edge effects using symplectic method similar to [3]. The value of EDGE_ORDER is ignored. Recommended for applications where symplecticity matters.
• 4: —- Edge effects using symplectic method developed by R. Lindberg based on K. Hwang’s method [45]. Note that there will be a trajectory offset when using this method that is particularly evident for small bending radii, due to extension of the fringe field outside the body of the magnet. To suppress this, adjustment of the FSE parameter can be performed automatically if FSE_CORRECTION is set to a non-zero value. If FSE_CORRECTION=1, the path-length is adjusted to match the nominal length, which is not physical; this behavior can be suppressed by setting FSE_CORRECTION=2. The EDGE_ORDER parameter is ignored.
• 5: Edge effects using symplectic method developed by R. Lindberg, similar to method 4, but with a more detailed treatment of the curved geometry and the presence of a transverse gradients. The EDGE_ORDER parameter may be used to include linear terms only (value of 1) or linear and higher-order terms (value of 2). The program curvedDipoleFringeCalc may be used to prepare the data needed for this method, which involves seven fringe integrals for each edge. These are input using the FRINGEnKm parameters, where n is 1 or 2 for the entrance or exit and m is 0 through 6.
• Other: — No edge effects.

Radiation effects

If SYNCH_RAD is non-zero, classical synchrotron radiation is included in tracking. Incoherent synchrotron radiation, when requested with ISR=1, normally uses gaussian distributions for the excitation of the electrons. (To exclude ISR for single-particle tracking, set ISR1PART=0.) Setting USE_RAD_DIST=1 invokes a more sophisticated algorithm that uses correct statistics for the photon energy and number distributions. In addition, if USE_RAD_DIST=1 one may also set ADD_OPENING_ANGLE=1, which includes the photon angular distribution when computing the effect on the emitting electron.

If TRACKING_MATRIX and SYNCH_RAD are non-zero, classical synchrotron radiation can be included in the ordinary matrix (e.g., for twiss_output and matrix_output) by setting SR_IN_ORDINARY_MATRIX to a non-zero value. Symplecticity is not assured, but the results may be interesting nonetheless. A more rigorous approach is to use moments_output. SR_IN_ORDINARY_MATRIX does not affect tracking.

Adding errors

When adding errors, care should be taken to choose the right parameters. The FSE and ETILT parameters are used for assigning errors to the strength and alignment relative to the ideal values given by ANGLE and TILT. One can also assign errors to ANGLE and TILT, but this has a different meaning: in this case, one is assigning errors to the survey itself. The reference beam path changes, so there is no orbit/trajectory error. The most common thing is to assign errors to FSE and ETILT. Note that when adding errors to FSE, the error is assumed to come from the power supply, which means that multipole strengths also change.

There are three modes for implementing alignment errors. Which is used is controlled by the value of the MALIGN_METHOD parameter:

• MALIGN_METHOD=0 — This selects the original method, which was the only one available before version 2021.1. The misalignment is referenced to the entrance face. The EYAW and EPITCH parameters are ignored.
• MALIGN_METHOD=1 — This selects a linearized method based on M. Venturini’s work [58], with misalignment referenced to the entrance face. The EYAW and EPITCH parameters are implemented.
• MALIGN_METHOD=2 — This selects a linearized method based on M. Venturini’s work [58], with misalignment referenced to the magnet center. The EYAW and EPITCH parameters are implemented.
• MALIGN_METHOD=3 — This selects an exact method based on M. Venturini’s work [58], with misalignment referenced to the entrance face. The EYAW and EPITCH parameters are implemented.
• MALIGN_METHOD=4 — This selects an exact method based on M. Venturini’s work [58], with misalignment referenced to the magnet center. The EYAW and EPITCH parameters are implemented.

For elements with non-zero TILT, error displacements and rotations are performed in the lab frame. An exception is the CCBEND, where error displacements and rotations are performed in the tilted frame.

Splitting dipoles

When dipoles are long, it is common to want to split them into several pieces, to get a better look at the interior optics. When doing this, care must be exercised not to change the optics. elegant has some special features that are designed to reduce or manage potential problems. At issue is the need to turn off edge effects between the portions of the same dipole.

First, one can simply use the divide_elements command to set up the splitting. Using this command, elegant takes care of everything.

Second, one can use a series of dipoles with the same name. In this case, elegant automatically turns off interior edge effects. This is true when the dipole elements directly follow one another or are separated by a MARK element.

Third, one can use a series of dipoles with different names. In this case, one must also use the EDGE1_EFFECTS and EDGE2_EFFECTS parameters to turn off interior edge effects.