About
The "tilts and decenters" table enables to enter the parameters related to the rotations and decenters to apply to a surface once another item than "no" is selected in the relevant "tilt-decenter" cell of the "mandatory parameters" table.

Except for the object surface and the first surface of the system, the "tilt-decenter" cell proposes 4 options:

- "no",
- "full user defined",
- "rel. to a surf.",
- "rel. to chief ray",
- "dec. rel. to a chief ray - tilt rel. to surf."

This cell is disabled for the object surface while only "no" and "full user defined" options are proposed for surface '1'.

"full user defined" allows to position a surface regarding the one just before.

"rel. to a surf." allows to position a surface regarding any preceeeding surface except the one just before.

"rel. to a chief ray." allows to position a surface regarding the chief ray.

"dec. rel. to a chief ray - tilt rel. to surf." allows to position the center of a surface regarding the incident chief ray and orientate it regarding a surface.


Option "Full user defined"

Let consider a surface "k" to be transformed, "p" being the identifier of the surface just before."p" is obviously "k-1" for any surface which number is larger than '1 ' and 'obj ' for surface number '1 '. All the cells of the "tilts and decenters" table related to the lines "aft. p" and "k" are then activated, "aft. p" being an abbreviation of after surface "p") .

These lines include the parameters tetax, tetay, tetaz, decx, decx, decy, Ptetax, Ptetay, Ptetaz, Pdecx and Pdecy.

tetax, tetay and tetaz are rotations repectively around the X, Y and Z axis.

decx and decy are translations respectively along the X and Y axis.

Ptetax, Ptetay, Ptetaz, Pdecx and Pdecy are the priorities for each transformation. They are integers from "1" to "5", "1" corresponding to the first transformation to process and "5" to the last one.

Each transformation is realized from the coordinate system obtained with the previous one.

The transformations defined on line "aft. p" are processed first, starting from the coordinate system of the surface "p". A translation with an amplitude corresponding to the distance "d" defined on line "p" of the "mandatory parameters" table is then applied along the Z axis obtained after this first set of transformations. Finally, the transformations defined on the line "k" of the "tilts and decenters" table are processed.

Note that the angles are defined in degrees and the translations in mm.
Note also that the object surface (identifier "obj") can not be transformed.

For more details concerning the "tilts and decenters" table, see the related tutorial.


Option "rel. to a surf."

This option can apply only for surfaces after the first surface of the system.

When selected for surface "k" in the "tilts-decenter" cell of the "mandatory parameters" table, all cells of line "k" in the "tilts and decenters" table are then activated including the "ref. surf." cell.

The reference surface is to be selected in the "ref. surf." cell displaying an option list of surfaces preceeding surface "k". Let call "ref" the surface selected in this list.

As above, let note "p" the number of the surface just before surface "k". Note that contrary to the "full user defined" option, all the cells of line "aft. p" in the "tilt and decenters" table are disactivated.

The coordinate system of surface "k" is preliminary positionned so that its XYZ axis coincide with those of surface "ref" and its origin coincide with the center of surface "p". A translation along the Z axis of surface "p" is first applied. The amplitude of this translation corresponds to the distance defined in cell "d" of line "p" in the "mandatory parameters" table.
The tilts and decenters defined on line "k" of the "tilts and decenters" table apply then with the same rules than those described above.

Therefore, if any surface in between surface "ref" and surface "k" is moved, that has no effect on the orientation of surface "k" nor on the coordinates of its center in the the XY plane of surface "ref". However, any motion of a surface in between can change the Z coordinate of surface "k" in the coordinate system of surface "ref".

Note that contrary to the "full user defined" option, decenters must be grouped either before or after the tilts.


Option "rel. to chief ray"

This option can apply only for surfaces after the first surface of the system.

When selected for surface "k", all cells of line "k" in the "tilts and decenters" table are then activated except the "ref. surf." cell.

Note that as for the "rel. to a surf." option, all the cells of line "aft. p" in the "tilt and decenters" table are disactivated.

The chief ray before surface "k" is first calculated and the coordinate system of surface "p" (just before surface "k") is translated in its XY plane so that its origin is moved on the chief ray. Tilts around X and Y are then processed so that the Z axis merges with the chief ray.

A translation along the new Z axis is then applied. The amplitude of this translation corresponds to the distance defined in cell "d" of line "p" in the "mandatory parameters" table.

Decenters in the XY plane are then applied to position the aperture center of surface "k" on the chief ray. If no aperture is defined or if the aperture is centered, obviously, no decenter is processed.

Finally, the tilts and decenters defined on line "k" of the "tilts and decenters" table apply with the same rules than those described above for positionning the coordinate system of surface "k" centered on its aperture relatively to the chief ray.

"Rel. to chief ray" option is very useful for off-axis systems using for instance off-axis mirrors, gratings, prisms, etc. This allows to enter a complex system with the certainty that the chief ray passes through the aperture of the components.

Note that the aperture must be defined by a surface before surface "k" or the system must be telecentric in the object space otherwise, the chief ray preceeding surface "k" cannot be calculated.

Note also that as for the "rel. to a surf." option, decenters must be grouped either before or after the tilts.


Option "dec. rel. to a chief ray - tilt rel. to surf."

This option can apply only for surfaces after the first surface of the system.

When selected for surface "k", all cells of line "k" in the "tilts and decenters" table are then activated including the "ref. surf." cell.

The reference surface is to be selected in the "ref. surf." cell displaying an option list of surfaces preceeding surface "k". Let call "ref" the surface selected in this list. Note that as for the "rel. to a surf." and "rel. to chief ray" options, all the cells of line "aft. p" in the "tilt and decenters" table are disactivated.

The chief ray before surface "k" is first calculated and the coordinate system of surface "p" (just before surface "k") is translated in its XY plane so that its origin is moved on the chief ray. Tilts around X and Y are then processed so that the Z axis merges with the chief ray.

A translation along the new Z axis is then applied. The amplitude of this translation corresponds to the distance defined in cell "d" of line "p" in the "mandatory parameters" table.

Decenters in the XY plane are then applied to position the aperture center of surface "k" on the chief ray. If no aperture is defined or if the aperture is centered, obviously, no decenter is processed.

Finally, the decenters defined on line "k" of the "tilts and decenters" table apply for positionning the aperture center of surface "k" relatively to the chief ray and the XYZ axis of surface "k" are tilted relatively to the surface "ref" accordingly to the tilts defined (on line "k") in the "tilts and decenters" table.

"Dec. rel. to a chief ray - tilt rel. to surf." option is very useful for off-axis systems using for instance off-axis mirrors, gratings, prisms, etc. positionned symetrically. This adds flexibility for entering a complex system with the certainty that the chief ray passes through the aperture of the components.

Note that the aperture must be defined by a surface before surface "k" or the system must be telecentric in the object space otherwise, the chief ray preceeding surface "k" cannot be calculated.

Note also that as for the "rel. to a surf." and "rel. to chief ray" options, decenters must be grouped either before or after the tilts.


For more details concerning the "tilts and decenters" table, see the related tutorial.