We now derive the relationship between a directional offset (as measured optically near the centres of each quadrant) and the corrective offset (as applies in the vicinity of the capacitors and PZTs). This is necessary because the offsets for a tilted plate are measured and corrected in two different locations, namely, the centre and edge of the plate respectively.

Suppose the upper plate is tilted along the plane

relative to the lower plate at

The location of emission-lines (such as those in Fig. 3)
determines the effective plate separation over that region.
The *effective* plate separation is the
volume of space between the plates divided by the cross-sectional
beam area isolated by the quadrant mask. Integrating over each
quadrant in turn gives effective plate separations

where and have values according to what region the quadrant occupies of the (

The problem is much simplified if the quadrant masks are oriented
such that the edges are parallel with the *x*-*y* axes. This is how
our system is operated in practice, deliberately decoupling the *XY*tilt motions. In the absence of rotation, Eqn. (4) reduces
to

with according to quadrant but where the

Now consider any two quadrants adjacent in the

Figure 4 shows a side-view along the *x*-direction of
an upper plate (*UU*') tilted relative to a lower one (*x*-axis).
*L*_{(-1,q)} and
*L*_{(+1,q)} are the effective plate separations
measured in each quadrant while
is the difference
between the two. Without loss of generality we set the *reference*
quadrant to be on the negative side and the *X* quadrant on the positive.
The offset
is the amount by which the plate needs
correction at the radius of the PZTs and capacitors, *R*_{c}.
From Fig. 4 we know through geometrical argument that

We showed in Fig. 3 how an offset such as can be measured directly by the offset of lines (panels

Substituting the radii of the
pupil plane beam (
*R*_{p} = 37.75 mm) and PZTs (*R*_{c} = 90 mm)
into Eqn. (7) yields

This is the relationship we need between measured offset, , and applied (corrective) offset, . It means that any measured offset in the

The precision of the technique is limited to the smallest steps by which the plates can be adjusted, not the smallest measurable deviation. By Eqn. (1), the smallest movement is a software step of 1, equivalent to nm or 0.01 % of our smallest plate spacing. This we can detect through motions as small as 0.09 nm near the plate centre. At the longest wavelengths this represents . This is much less than the parallelism criterion, over the full range of TTF wavelengths.