Non-meridional 2D motion.

CPO2D systems with axial symmetry can also be used for non-meridional rays. Meridional rays start by moving in a plane that contains the axis, and continue to move in that plane, perhaps crossing the axis one or more times. On the other hand non-meridional rays start at an angle to the initial plane or are moved transversely by a magnetic field. They therefore have a non-zero angular momentum about the axis and so cannot pass through the axis. Similarly for CPO2D systems with planar symmetry can also be used for non-xz-plane rays -that is, rays that do not have zero velocity in the y direction.

Examples of non-meridional motion are given in test2d22, non-meridional rays, cylindrical symmetry; test2d23, non-xz-plane rays, planar symmetry; test2d28, helical motion in a uniform magnetic field; test2d29, magnetic Glaser lens; xmpl2d51, non-meridional beam in a uniform magnetic field; xmpl2d52, repeated non-meridional space-charge iterations.

In the non-meridional mode there is a choice of viewing the final ray plots either as r versus z or as r*cos(phi) versus z. This choice appears in the ‘View’ drop-down menu. Here phi is the angle around the axis, so therefore r*cos(phi) is essentially the projection of the motion on the fixed xz plane. The value of phi always starts as 0, but r*cos(phi) can of course become negative. On the other hand the plot of r versus z shows the maximum distance r from the axis (and r is of course always >= 0).

It should be remembered in axially symmetric systems a ‘ray’ seen in the plots actually represent a cut through an axially symmetric sheet. The plot of r versus z is therefore the more realistic one, and is the only one allowed for space-charge plots.

There is also an option (in databuilder/printing levels/rays) to output ray step information as x,y,z,vx,vy,vz,etc. This is intended to help in understanding the progress of a trajectory.

The conversion from the xyz coordinates to the r,phi,z coordinates is:

r = sqrt(x^2 + y^2), phi = atan(y/x), z = z.

The conversion from the r,phi,z coordinates to the xyz coordinates is:

x = r(cos(phi), y = r*sin(phi), z = z.

This xyz option is available only for the ray steps, and is not applied to any other output information, such as test plane data.

The non-meridional option cannot be combined with the options for entering total energy, or using variable mass, time-dependence, use-saved-trajectories or focus iterations.

The program allows the option to be used for space-charge simulations, but users must remember that the space-harge has rotational symmetry. The option is therefore only suitable for space-charge sources that are discs or rings that are centred on the axis.

Examples are given in test2d22.dat and test2d23.dat.

As a simpler example of non-meridional motion in a system that has axial symmetry, we can consider a field-free system. Here it is helpful to convert the (r,z,phi) coordinates to (x,y,z), using x = rcos(phi), y = rsin(phi), so that the 3-dimensional nature of the motion can be followed more easily.

In this example the ray starts at z = 0 and finishes at z = 10. The velocity in the x direction is twice the velocity in the z direction. The ray starts with x = 1 and its velocity in the y direction is either 0 (meridional) or non-zero (non-meridional). The table shows the resulting final coordinates.

Initial Final

r z vr vz vphi x y z vx vy vz  x y z r phi

1 0 2 1 0 1 0 0 2 0 1  21  0 10 21 0

1 0 2 1 1 1 0 0 2 1 1  21  10 10 23.26 0.444

For systems that have axial symmetry non-meridional motion can only be simulated by starting with y = 0, phi = 0 and with non-zero values of vy. If the practical system includes a ray that has non-zero values of y and phi and non-zero values of vx and/or vy then the ray should be rotated about the axis so that it initially has phi = 0 and then the output coordinates should be rotated back again to give the final coordinates.

For users who are editing or constructing an 'input data file' without the use of the data-builder -that is, pre-processor:

But Manual editing is certainly not recommended -it is a relic from the time when the databuilder was not available All users are strongly encouraged to use the databuilder, which always gives the correct formats and which has many options for which the formats are not described or easily deduced.

The line 'set of single rays' should be replaced by 'non meridional set of single rays' (or more simply, 'non').

The value of vphi should be inserted after the value vz.