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In this chapter details of the different types of field maps in OPAL-t are presented. OPAL-t can use many different types of field maps input in several different file formats. What types of maps are supported and in what format has tended to be a function mostly of what developers have needed, and to a lesser extent what users have asked for. The list below shows all field maps that are currently supported and also field maps that are not yet supported, but on the list of things to do when we get a chance.

The possibility to add comments (almost) everywhere in field map files is common to all field maps. Comments are initiated by a # and contain the rest of a line. Comments are accepted at the beginning of the file, between the lines and at the end of a line. If in the following sections two values are shown on one line then they have to be on the same line. They should not be separated by a comment and, consequently, be on different lines. Three examples of valid comments:

The following examples will break the parsing of the field maps:

All field maps that are in ASCII are normalized with the maximum absolute value of the longitudinal field on the axis. The only exceptions is the type 1DProfile1. This behavior can be disabled by adding FALSE at the end of the first line of the field map.

If OPAL-t encounters an error while parsing a field map it disables the corresponding element, outputs a warning message and continues the simulation. The following messages may be output:

In this example there is something wrong with the number of grid spacings provided in the header of the file. Make sure that you provide the number of grid spacings and not the number of grid points! The two numbers always differ by 1.

Again there seems to be something wrong with the number of grid spacings provided in the header. In this example OPAL-t found more lines than it expected. Note that comments and empty lines at the end of a file are ignored such that they don’t cause this warning.

Where “_error_msg_” is either

“_expecting_” is replaced by the types of values OPAL-t expects on the line. E.g. it could be replaced by double double int. Finally “_found_” is replaced by the actual content of the line without any comment possibly following the values. If line 3 of a file consists of -60.0 60.0 # This is an other invalid comment # 9999 OPAL-t will output -60.0 60.0.

This warning could be issued if the file name is mistyped or otherwise if the file couldn’t be read.

In this case OPAL-t didn’t recognize the string of characters which identify the type of field map stored in the file.

For one-dimensional field maps an other warning may be issued:

This warning is issued when the low pass filter that is applied to the field sampling uses too few Fourier coefficients. In this case increase the number of Fourier coefficients, see the next section for details. The relevant criteria are that

and

where F_{z_i} is the field sampling as in the file and \tilde{F}_{z,i} is the one-dimensional field reconstructed from the result received after applying the low pass filter.

Field maps in OPAL-t come in three basic types:

1. 2D or 3D field map. For this type of map, a field is specified on a grid and linear interpolation is used to find field values at intermediate points.

2. 1D on axis field map. For this type of map, one on-axis field profile is specified. OPAL-t calculates a Fourier series from this profile and then uses the first, second and third derivatives of the series to compute the off-axis field values. (This type of field is very smooth numerically, but can be inaccurate far from the field axis.) Only a few (user specified) terms from the Fourier series are used.

3. Enge function [enge] field map. This type of field map uses Enge functions to describe the fringe fields of a magnet. Currently, this is only used for RBEND and SBEND elements see Section [RBend,SBend].

It is important to note that in all cases, the input field map will be normalized so that the peak field magnitude value on axis is equal to either 1 MV/m in the case of electric field maps (static or dynamic), or 1 T in the case of magnetic field maps. (The sign of the values from the field map are preserved.) Therefore, the field multiplier for the map in your simulation will be the peak field value on axis in those respective units.

Depending on the field map type, OPAL-t uses different length units (either cm or meters). This is due to the origin programs of the field maps used (e.g. Poisson/Superfish [superfish] uses cm). So be careful.

There are no required field extensions for any OPAL-t field map (e.g. .T7, .dat etc.). OPAL-t determines the type of field map by a string descriptor which is the first element on the first line of the file. Below we list the possible descriptors. (Note that we list all of the descriptors/field map types that we plan to eventually implement. Not all of them are, which is indicated in the description.)

1DElectroStatic

1D electrostatic field map. 1D field maps are described by the on-axis field.
Not implemented yet. A work around is to use a 1DDynamic field map with a very low frequency.

1DMagnetoStatic

1D magnetostatic field map. See Section [1DMagnetoStatic].

AstraMagnetoStatic

1D magnetostatic field map with possibly non-equidistant sampling. This file type is compatible with ASTRA field maps with small changes. See Section [ AstraMagnetostatic].

1DDynamic

1D dynamic electromagnetic field map. See Section [1DDynamic].

AstraDynamic

1D dynamic electromagnetic field map with possibly non-equidistant sampling. This file type is compatible with ASTRA field maps with small changes. See Section [AstraDynamic].

1DProfile1

This type of field map specifies the Enge functions (see [enge]) for the entrance and exit fringe fields of a magnet. Currently this type of field map is only used by RBEND and SBEND elements see Section [RBend,SBend]. See Section [1DProfile1].

2DElectroStatic

2D electrostatic field map. 2D field maps are described by the electromagnetic field in one half-plane. See Section [2DElectroStatic].

2DMagnetoStatic

2D magnetostatic field map. Other than this descriptor at the head of the file, the format for this field map type is identical to the T7 file format as produced by Poisson [superfish]. See Section [2DMagnetoStatic].

2DDynamic

2D dynamic electromagnetic field map. Other than this descriptor at the head of the file, the format for this field map type is identical to the T7 field format as produced by Superfish [superfish]. See Section [2DDynamic].

3DElectroStatic

3D electrostatic field map.
Not implemented yet.

3DMagnetoStatic

3D magnetostatic field map, see Section [3DMagnetoStatic].

3DMagnetoStatic_Extended

2D magnetostatic field map of field on mid-plane that OPAL extends to 3D.

3DDynamic

3D dynamic electromagnetic field map. See Section [3DDynamic].

We will give examples and descriptions of each of the implemented field map types in the sections below.

In the case of 2D and 3D field maps an additional string has to be provided describing the orientation of the field map.

For 2D field maps this can either be

XZ

if the primary direction is in z direction and the secondary in r direction.

ZX

if the primary direction is in r direction and the secondary in z direction.

For 3D field maps this can be

XYZ

if the primary direction is in z direction, the secondary in y direction and the tertiary in x direction

Each line after the header corresponds to a grid point of the field map. This point can be referred to by two indices in the case of a 2D field map and three indices in the case of a 3D field map, respectively. Each column describes either E_z,\; E_r,\; B_z,\; B_r\; \text{or}\;H_{\phi} in the 2D case and E_x\;, E_y\;, E_z,\; B_x,\; B_y,\;B_z in the 3D case.

By primary, secondary and tertiary direction is meant the following (see also Figure 1):

• The index of the primary direction increases the fastest, the index of the tertiary direction the slowest.

• The order of the columns is accordingly: if the z direction in an 2D electrostatic field map is the primary direction then E_z is on the first column, E_r on the second. For all other cases it’s analogous.

• For the 2D dynamic case in XZ orientation there are four columns: E_z, E_r, |E| (unused) and H_{\phi} in that order. In the other orientations the first and the second columns are interchanged ,but the third and fourth columns are unchanged.

For some 1D field maps, there exists a Boolean attribute, FAST, which can be used to speed up the calculation. When set to true (FAST = TRUE), OPAL-t will generate a 2D internal field map and then use bi-linear interpolation to calculate field values during the simulation, rather than the generally slower Fourier coefficient technique. The caution here is that this can introduce unwanted numerical noise if you set the grid spacing too coarse for the 2D map.

As a general warning: be wise when you choose the type of field map to be used! Figure [fieldnoiseexample] shows three pictures of the longitudinal phase space after three gun simulations using different types of field maps. In the first picture we used a 1DDynamic field map see Section [1DDynamic] resulting in a smooth longitudinal distribution. In the middle picture we set the FAST attribute to true, resulting in some fine structure in the phase space due to the bi-linear interpolation of the internally generated 2D field map. Finally, in the last figure, we generated directly a 2D field map from Superfish [superfish]. Here we could observe two different structures: first the fine structure, stemming from the bi-linear interpolation, and secondly a much stronger structure of unknown origin, but presumably due to errors in the Superfish [superfish] interpolation algorithm.

A 1D field map describing a magnetostatic field using 10000 grid points (9999 grid spacings) in the longitudinal direction. The field is non-negligible from -60.0cm to +60.0cm relative to ELEMEDGE in the longitudinal direction. From the 10000 field values, 5000 complex Fourier coefficients are calculated. However, only 40 are kept when calculating field values during a simulation. OPAL-t normalizes the field values internally such that \max(|B_{\text{on axis}}|) = {1.0}{T}. If the FAST attribute is set to true in the input deck, a 2D field map is generated internally with 200 values in the radial direction, from 0cm to 2cm, for each longitudinal grid point.

A 1DMagnetoStatic field map has the general form shown in Table [1DMagnetoStatic]. The first three lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (1DMagnetoStatic) and how many Fourier coefficients to keep (N_{Fourier}) when doing field calculations.

Line 2

This tells gives the extent of the field map (from z_{start} to z_{end}) relative to the ELEMEDGE of the field map, and how many grid spacings there are in the field map.

Line 3

If one sets FAST = TRUE for the field map, this tells OPAL-t the radial extent of the internally generated 2D field map. Otherwise this line is ignored. (Although it must always be present.)

The lines following the header give the 1D field map grid values from 1 to N_{z} + 1. From these, N_{z}/2 complex Fourier coefficients are calculated, of which only N_{Fourier} are used when finding field values during the simulation.

Figure [1DMagnetoStatic] gives an example of a 1DMagnetoStatic field file.

A 1D field map describing a magnetostatic field using N non-equidistant grid points in the longitudinal direction. From these values N equidistant field values are computed from which in turn N/2 complex Fourier coefficients are calculated. In this example only 40 Fourier coefficients are kept when calculating field values during a simulation. The z-position of each field sampling is in the first column (in meters), the corresponding longitudinal on-axis magnetic field amplitude is in the second column. As with the 1DMagnetoStatic see Section [1DMagnetoStatic] field maps, OPAL-t normalizes the field values to \max(|B_{\text{on axis}}|) = {1.0}{T}. In the header only the first line is needed since the information on the longitudinal dimension is contained in the first column of the data. (OPAL-t does not provide a FAST version of this map type.)

An AstraMagnetoStatic field map has the general form shown in Table [AstraMagnetoStatic]. The first line forms the file header and tells OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (AstraMagnetoStatic) and how many Fourier coefficients to keep (N_{Fourier}) when doing field calculations.

The lines following the header gives N non-equidistant field values and their corresponding z positions (relative to ELEMEDGE). From these, OPAL-t will use cubic spline interpolation to find N equidistant field values within the range defined by the z positions. From these equidistant field values, N/2 complex Fourier coefficients are calculated, of which only N_{Fourier} are used when finding field values during the simulation.

Figure [AstraMagnetoStatic] gives an example of an AstraMagnetoStatic field file.

A 1D field map describing a dynamic field using 5000 grid points (4999 grid spacings) in the longitudinal direction. The field is non-negligible from -3.0cm to 57.0cm relative to ELEMEDGE in the longitudinal direction. The field frequency is 1498.953425154MHz. From the 5000 field values, 2500 complex Fourier coefficients are calculated. However, only 40 are kept when calculating field values during the simulation. OPAL-t normalizes the field values internally such that max(|E_{on axis}|) = {1}{MV/m}. If the FAST switch is set to true in the input deck, a 2D field map is generated internally with 200 values in the radial direction, from 0.0cm to 2.0cm, for each longitudinal grid point.

A 1DDynamic field map has the general form shown in Table [1DDynamic]. The first four lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (1DDynamic) and how many Fourier coefficients to keep (N_{Fourier}) when doing field calculations.

Line 2

This tells gives the extent of the field map (from z_{start} to z_{end}) relative to the ELEMEDGE of the field map, and how many grid spacings there are in the field map.

Line 3

Field frequency.

Line 4

If one sets FAST = TRUE for the field map, this tells OPAL-t the radial extent of the internally generated 2D field map. Otherwise this line is ignored. (Although it must always be present.)

The lines following the header give the 1D field map grid values from 1 to N_{z} + 1. From these, N_{z}/2 complex Fourier coefficients are calculated, of which only N_{Fourier} are used when finding field values during the simulation.

Figure [1DDynamic] gives an example of a 1DDynamic field file.

A 1D field map describing a dynamic field using N non-equidistant grid points in longitudinal direction. From these N non-equidistant field values N equidistant field values are computed from which in turn N/2 complex Fourier coefficients are calculated. In this example only 40 Fourier coefficients are kept when calculating field values during the simulation. The z-position of each sampling is in the first column (in meters), the corresponding longitudinal on-axis electric field amplitude is in the second column. OPAL-t normalizes the field values such that \max(|E_{\text{on axis}}|) = {1}{MV/m}. The frequency of this field is 2997.924MHz. (OPAL-t does not provide a FAST version of this map type.)

An AstraDynamic field map has the general form shown in Table [AstraDynamic]. The first line forms the file header and tells OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (AstraDynamic) and how many Fourier coefficients to keep (N_{Fourier}) when doing field calculations.

Line 2

Field frequency.

The lines following the header gives N non-equidistant field values and their corresponding z positions (relative to ELEMEDGE). From these, OPAL-t will use cubic spline interpolation to find N equidistant field values within the range defined by the z positions. From these equidistant field values, N/2 complex Fourier coefficients are calculated, of which only N_{Fourier} are used when finding field values during the simulation.

Figure [AstraDynamic] gives an example of an AstraDynamic field file.

A 1DProfile1 field map is used to define Enge functions [enge] that describe the fringe fields for the entrance and exit of a magnet:

where D is the full gap of the magnet, N_{order} is the Enge function order and z is the distance from the Enge function origin perpendicular to the edge of the magnet. The constants, c_n, and the Enge function origin are fitted parameters chosen to best represent the fringe field of the magnet being modeled.

A 1DProfile1 field map describes two Enge functions: one for the magnet entrance and one for the magnet exit. An illustration of this is shown in Figure 3. In the top part of the figure we see a plot of the relative magnet field strength along the mid-plane for a rectangular dipole magnet. To describe this field with a 1DProfile1 field map, an Enge function is fit to the entrance fringe field between zbegin_entry and zend_entry in the figure, using the indicated entrance origin. Likewise, an Enge function is fit to the exit fringe field between zbegin_exit and zend_exit using the indicated exit origin. The parameters for these two Enge functions are subsequently entered into a 1DProfile1 field map, as described below.

Currently, 1DProfile1 field maps are only implemented for RBEND and SBEND elements see Section [RBend,SBend,opaltrbendsbendfields].

A 1DProfile1 field map has the general form shown in Table [1DProfile1]. The first three lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (1DProfile1), the Enge coefficient order for the entrance fringe fields (N_{Enge\,Entrance}), the Enge coefficient order for the exit fringe fields (N_{Enge\,Exit}), and the gap of the magnet.

Line 2

The first three values on the second line are used to define the extent of the fringe fields for the entrance region of the magnet. This can be done two different ways as will be described below see Section [1DProfile1Type1,1DProfile1Type2]. The fourth value on line 2 is not currently used (but must still be present).

Line 3

The first three values on the third line are used to define the extent of the fringe fields for the exit region of the magnet. This can be done two different ways as will be described below see Section [1DProfile1Type1,1DProfile1Type2]. The fourth value on line 3 is not currently used (but must still be present).

The lines following the three header lines give the entrance region Enge coefficients from c_0 to c_{N_{Enge\,Entrance}}, followed by the exit region Enge coefficients from c_0 to c_{N_{Enge\,Exit}}.

There are two types of 1DProfile1 field map files: 1DProfile Type 1 and 1DProfile1 Type 2. The difference between the two is a small change in how the entrance and exit fringe field regions are described. This will be explained in Section [1DProfile1Type1] and Section [1DProfile1Type2].

1.12.1. 1DProfile1 Type 1 for Bend Magnet

\begin{aligned}
  Entrance\,Parameter\,1 &= Entrance\,Parameter\,2 - \Delta_{1} \\
  Entrance\,Parameter\,3 &= Entrance\,Parameter\,2 + \Delta_{2} \\
  Exit\,Parameter\,2 &= L - Entrance\,Parameter\,2 \\
  Exit\,Parameter\,1 &= Exit\,Parameter\,2 - \Delta_{3} \\
  Exit\,Parameter\,3 &= Exit\,Parameter\,2 + \Delta_{4}\end{aligned}

\begin{aligned}
L &= Exit\,Parameter\,2 - Entrance\,Parameter\,2 \\
\Delta_{1} &= Entrance\,Parameter\,2 - Entrance\,Parameter\,1 \\
\Delta_{2} &= Entrance\,Parameter\,3 - Entrance\,Parameter\,2 \\
\Delta_{3} &= Exit\,Parameter\,2 - Exit\,Parameter\,1 \\
\Delta_{4} &= Exit\,Parameter\,3 - Exit\,Parameter\,2\end{aligned}

1DProfile1 6 7 3.0
-6.0 -2.0 2.0  1000
24.0 28.0 32.0 0
  0.00000e+00
  4.36222e-06
  8.83270e-06
  + 9 lines
  1.32490e-05
  1.73710e-05
  2.18598e-05

1.12.2. 1DProfile1 Type 2 for Bend Magnet

\begin{aligned}
  Entrance\,Parameter\,2 &= \perp \text{distance of entrance Enge function origin from magnet entrance edge} \\
  Exit\,Parameter\,2 &= \perp \text{distance of exit Enge function origin from magnet exit edge}\end{aligned}

\begin{aligned}
  Entrance\,Parameter\,1 &= Entrance\,Parameter\,2 - \Delta_{1} \\
  Entrance\,Parameter\,3 &= Entrance\,Parameter\,2 + \Delta_{2} \\
  Exit\,Parameter\,1 &= Exit\,Parameter\,2 - \Delta_{3} \\
  Exit\,Parameter\,3 &= Exit\,Parameter\,2 + \Delta_{4}\end{aligned}

\begin{aligned}
\Delta_{1} &= Entrance\,Parameter\,2 - Entrance\,Parameter\,1 \\
\Delta_{2} &= Entrance\,Parameter\,3 - Entrance\,Parameter\,2 \\
\Delta_{3} &= Exit\,Parameter\,2 - Exit\,Parameter\,1 \\
\Delta_{4} &= Exit\,Parameter\,3 - Exit\,Parameter\,2\end{aligned}

1DProfile1 6 7 3.0
-6.0 -2.0 2.0 0
-2.0  2.0 6.0 0
  0.00000e+00
  4.36222e-06
  8.83270e-06
  + 9 lines
  1.32490e-05
  1.73710e-05
  2.18598e-05

A 2D field map describing an electrostatic field using 5000 grid points in the longitudinal direction times 200 grid points in the radial direction. The field between the grid points is calculated using bi-linear interpolation. The field is non-negligible from -3.0cm to 51.0cm relative to ELEMEDGE and the 200 grid points in the radial direction span the distance from 0.0cm to 2.0cm. The field values are ordered in XZ orientation, so the index in the longitudinal direction changes fastest and therefore E_z values are stored in the first column and E_r values in the second see Section [fieldorientation]. OPAL-t normalizes the field so that max(|E_{z, \text{ on axis}}|) = {1}{MVm^{-1}}.

A 2DElectroStatic field map has the general form shown in Table [2DElectroStatic]. The first three lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (2DElectroStatic) and the field orientation see Section [fieldorientation].

Line 2

This gives the extent of the field map and how many grid spacings there are in the fastest changing index direction see Section [fieldorientation].

Line 3

This gives the extent of the field map and how many grid spacings there are in the slowest changing index direction (see Section [fieldorientation].

The lines following the header give the 2D field map grid values from 1 to N = (N_{z} + 1) \times (N_{r} + 1). The order of these depend on the field orientation see Section [fieldorientation] and can be one of two formats:

If Orientation = XZ

E_{z} (MV/m) E_{r} (MV/m)

If Orientation = ZX

E_{r} (MV/m) E_{z} (MV/m)

Figure [2DElectroStatic] gives an example of a 2DElectroStatic field file.

A 2D field map describing a magnetostatic field using 5000 grid points in the longitudinal direction times 200 grid points in the radial direction. The field between the grid points is calculated using bi-linear interpolation. The field is non-negligible from -3.0cm to 51.0cm relative to ELEMEDGE and the 200 grid points in the radial direction span the distance from 0.0cm to 2.0cm. The field values are ordered in the ZX orientation, so the index in the radial direction changes fastest and therefore B_r values are stored in the first column and B_z values in the second see Section [fieldorientation]. OPAL-t normalizes the field so that max(|B_{z,\text{ on axis}}|) = {1}{T}.

A 2MagnetoStatic field map has the general form shown in Table [2DMagnetoStatic]. The first three lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (2DMagnetoStatic) and the field orientation see Section [fieldorientation].

Line 2

This gives the extent of the field map and how many grid spacings there are in the fastest changing index direction see Section [fieldorientation].

Line 3

This gives the extent of the field map and how many grid spacings there are in the slowest changing index direction (see Section [fieldorientation].

The lines following the header give the 2D field map grid values from 1 to N = (N_{z} + 1) \times (N_{r} + 1). The order of these depend on the field orientation see Section [fieldorientation] and can be one of two formats:

If Orientation = XZ

B_{z} (T) B_{r} (T)

If Orientation = ZX

B_{r} (T) B_{z} (T)

Figure [2DMagnetoStatic] gives an example of a 2DMagnetoStatic field file.

A 2D field map describing a dynamic field oscillating with a frequency of 1498.953425154MHz. The field map provides 4122 grid points in the longitudinal direction times 76 grid points in radial direction. The field between the grid points is calculated with a bi-linear interpolation. The field is non-negligible between -3.0cm and 51.0cm relative to ELEMEDGE and the 76 grid points in radial direction span the distance from 0.0cm to 1.0cm. The field values are ordered in the XZ orientation, so the index in the longitudinal direction changes fastest and therefore E_z values are stored in the first column and E_r values in the second. The third column contains the electric field magnitude, |E|, and is not used (but must still be included). The fourth column is H_{\phi} in A/m. The third and fourth columns are always the same and do not depend on the field orientation see Section [fieldorientation]. OPAL-t normalizes the field so that max(|E_{z,\text{ on axis}}|) = {1}{MVm^{-1}}.

A 2DDynamic field map has the general form shown in Table [2DDynamic]. The first four lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (2DDynamic) and the field orientation see Section [fieldorientation].

Line 2

This gives the extent of the field map and how many grid spacings there are in the fastest changing index direction see Section [fieldorientation].

Line 3

Field frequency.

Line 4

This gives the extent of the field map and how many grid spacings there are in the slowest changing index direction see Section [fieldorientation].

The lines following the header give the 2D field map grid values from 1 to N= (N_{z} + 1) \times (N_{r} + 1). The order of these depend on the field orientation see Section [fieldorientation] and can be one of two formats:

If Orientation = XZ

E_{z} (MV/m) E_{r} (MV/m) |E| (MV/m) H_{\phi} (A/m)

If Orientation = ZX

E_{r} (MV/m) E_{z} (MV/m) |E| (MV/m) H_{\phi} (A/m)

The third item (the field magnitude) on each data line is not used by OPAL-t, but must be there.

Figure [2DDynamic] gives an example of a 2DDynamic field file.

A 3D field map describing a magnetostatic field. The field map provides 4122 grid points in z-direction times 228 grid points in x-direction and 152 grid points in y-direction. The field between the grid points is calculated with a tri-linear interpolation. The field is non-negligible between -3.0cm to 51.0cm relative to ELEMEDGE, the 228 grid points in x-direction range from -1.5cm to 1.5cm and the 152 grid points in y-direction range from -1.0cm to 1.0cm relative to the design path. The field values are ordered such that the index in z-direction changes fastest, then the index in y-direction while the index in x-direction changes slowest. The columns correspond to B_x, B_y and B_z.

A 3DMagnetoStatic field map has the general form shown in Table [3DMagnetoStatic]. The first five lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (3DMagnetoStatic).

Line 3

This gives the extent of the field map and how many grid spacings there are in the slowest changing index direction.

Line 4

This gives the extent of the field map and how many grid spacings there are in the next fastest changing index direction.

Line 5

This gives the extent of the field map and how many grid spacings there are in the fastest changing index direction.

The lines following the header give the 3D field map grid values from 1 to N= (N_{z} + 1) \times (N_{y} + 1) \times (N_{x} + 1). Figure [3DMagnetoStatic] gives an example of a 3DMagnetoStatic field file.

A 3D field map describing a magnetostatic field on the mid-plane. The field map provides 466 grid points in z-direction times 134 grid points in x-direction. The field is non-negligible between -22.425cm to 47.425cm relative to ELEMEDGE, the 134 grid points in x-direction range from -9.9254cm to 9.9254cm. The field should be integrated using Maxwell’s equations from the mid-plane to 2.0cm using 16 grid points. The mid-plane is regarded as a perfect magnetic conductor (PMC) i.e. the magnetic field on the mid-plane has no tangential component. This leads to a symmetry where the perpendicular component is mirrored whereas the tangential component is anti-parallel. Instead of integrating the field from the mid-plane to -2.0cm and 1.0cm we only integrate it to +2.0cm and store only the upper half of the field map. For positions R(x,\;-y,\;z) with y > 0.0 the correct field can then be derived from the R(x,\;y,\;z).

A 3DMagnetoStatic_Extended field map has the general form shown in Table [3DMagnetoStatic_Extended]. The first four lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (3DMagnetoStatic_Extended).

Line 2

This gives the extent of the field map and how many grid spacings there are in the slowest changing direction.

Line 3

This gives the extent of the field map and how many grid spacings there are in the next fastest changing direction.

Line 4

This gives the extent of the field map and how many grid spacings there are in the fastest changing direction.

The lines following the header give the 3D field map grid values from 1 to N= (N_{z} + 1) \times (N_{x} + 1). The order of these depend on the field orientation see Section [fieldorientation] and can currently only be the format shown in Table [3DMagnetoStatic_Extended].

Figure [3DMagnetoStatic_Extended] gives an example of a 3DMagnetoStatic_Extended field file.

A 3D field map describing a dynamic field oscillating with 1.4989534. The field map provides 4122 grid points in z-direction times 228 grid points in x-direction and 152 grid points in y-direction. The field between the grid points is calculated with a tri-linear interpolation. The field is non-negligible between -3.0cm to 51.0cm relative to ELEMEDGE, the 228 grid points in x-direction range from -1.5cm to 1.5cm and the 152 grid points in y-direction range from -1.0cm to 1.0cm relative to the design path. The field values are ordered such that the index in z-direction changes fastest, then the index in y-direction while the index in x-direction changes slowest. The columns correspond to E_x, E_y, E_z, H_x, H_y and H_z. OPAL-t normalizes the field so that max(|E_{z,\text{ on axis}}|) = {1}{MVm^{-1}}.

A 3DDynamic field map has the general form shown in Table [3DDynamic]. The first five lines form the file header and tell OPAL-t how the field map data is being presented:

Line 1

This tells OPAL-t what type of field file it is (3DDynamic).

Line 2

Field frequency.

Line 3

This gives the extent of the field map and how many grid spacings there are in the slowest changing index direction.

Line 4

This gives the extent of the field map and how many grid spacings there are in the next fastest changing index direction.

Line 5

This gives the extent of the field map and how many grid spacings there are in the fastest changing index direction.

The lines following the header give the 3D field map grid values from 1 to N= (N_{z} + 1) \times (N_{y} + 1) \times (N_{x} + 1). Figure [3DDynamic] gives an example of a 3DDynamic field file.