Deriving a Corrective Offset

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

 

z(x,y) = L₀ - Ax - By (3)

relative to the lower plate at z(x,y) = 0. Here, L₀ is the separation of the plate centres (in $\mu$m) and A and B are constants proportional to slope in x and y. We assume that the x-y axes align with the XY capacitors and that p-q axes define the four segments of the pupil plane masks. Initially we also assume the p-q axes to be rotated an angle $\phi$ counter-clockwise from the positive x-axis.

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

$$L_{(p,q)} = \frac {\int\!\!\int\!\!\int dV} {\int\!\!\int dA... ...}{3 \pi} \Bigl[ (pA+qB) \cos \phi + (-qA+pB) \sin \phi \Bigr],$$ (4)

where $(p,q) \in \{ (\pm 1, \pm 1) \}$ and have values according to what region the quadrant occupies of the (p,q)-plane (Fig. 4, inset). The beam radius, R_p, is defined by the radial size of the beam at the pupil plane (37.75 mm).

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 XYtilt motions. In the absence of rotation, Eqn. (4) reduces to

$$L_{(p,q)} = L_0 - \frac{4 R_p}{3 \pi} (pA+qB),$$ (5)

with $(p,q) \in \{ (\pm 1, \pm 1) \}$ according to quadrant but where the p-q and x-y axes are now aligned. Equating Eqs. (3) and (5) gives the linear locus of points across a quadrant (p,q) at which the plates are separated by exactly L_(p,q),

$$Ax + By = \frac{4 R_p}{3 \pi} (pA+qB) .$$ (6)

Now consider any two quadrants adjacent in the x direction (that is, with common q but opposite p values). It can be shown by Eqn. (6) that the separation (in x) between the L_(p,q)-loci of both quadrants remains fixed at $8 R_p /3 \pi$ for all values of y. In other words, the baseline separating L_(-1,q) and L_(+1,q) in the x-direction is a constant, irrespective of y. The same is true for loci separation in the y-direction along lines of constant x.

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 $\Delta Z_X$ 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 $\Delta Z'_X$ 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

$$\Delta Z'_X = \frac{3 \pi R_c}{8 R_p} \Delta Z_X .$$ (7)

We showed in Fig. 3 how an offset such as $\Delta Z_X$can be measured directly by the offset of lines (panels X and ref in the case of the x-offset).

Substituting the radii of the pupil plane beam ( R_p = 37.75 mm) and PZTs (R_c = 90 mm) into Eqn. (7) yields

$$\Delta Z'_X = 2.8 \Delta Z_X .$$ (8)

This is the relationship we need between measured offset, $\Delta Z_X$, and applied (corrective) offset, $\Delta Z'_X$. It means that any measured offset in the x direction must be corrected by 2.8 times that offset in the opposite direction. Similar arguments find an identical scale factor between $\Delta Z_Y$ and $\Delta Z'_Y$. The units in Eqn. (8) can either be the measured software control units or physical units ($\mu$m) through Eqn. (1).

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 $\Delta Z'_X = 0.24$ nm or $\sim$ 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 $\lambda/10000$. This is much less than the $\lambda/500$ parallelism criterion, over the full range of TTF wavelengths.