Astigmatism correction for the 2.5-m primary mirror: I. axial
supports
Sloan Digital Sky Survey Telescope Technical Note
19980805
Walter
Siegmund
Contents
Introduction
The SDSS 2.5-m primary mirror is supported in a manner similar to
the Apache Point Observatory 3.5-m telescope primary.1
Separate systems support the axial and transverse components of the
2.5-m mirror weight vector. The axial support is provided by 48 air
pistons. The pistons have the same diameter thereby applying equal
forces with the same air pressure. The 48 pistons are divided into
three groups of 16 each. Each group supports a 120° sector of
the mirror. The portion of the weight of the mirror that is not
supported by the air pistons is sensed by load cells located near the
center of each 120° sector. Each load cell controls the air
pressure to the pistons in its sector so that the force on the load
cell is about 10 N. This force is too small to cause significant
distortion of the mirror.
The transverse support is provided by 18 air pistons, supported by
cantilevers from the mirror cell, that act on the mirror local center
of gravity surface when the mirror is lowered into place. Each piston
pushes on a steel force spreader that, in turn, pushes on four
nickel-iron alloy blocks bonded to the mirror ribs. All transverse
air pistons are the same diameter. A single load cell senses the
unsupported mirror weight and controls the air pressure to the
transverse support pistons. The remaining rotational and lateral
degrees of freedom are constrained by two links to the mirror cell.
With the mirror pointed at the horizon, these are located near the
top and bottom edge of the mirror. They are attached to the back of
the mirror and act horizontally.
During figuring and testing, slurry skirts, pressure seals and
tangent rods were attached to the mirror to control contamination,
allow pressurization of the inside of the mirror to prevent tool
pressure-induced dimpling, and to constrain the mirror on its
supports. It is believed that that one of the seals or skirts applied
forces at one or more points along either the inside or outside edge
of the mirror. Upon removal, the mirror relaxed and a small amount of
astigmatism appeared. Testing by the optician2, after the
slurry skirts, pressure seals and tangent rods were removed,
indicated that the Zernike coefficient of surface astigmatism,
R22 was approximately 230 nm
(R22(r/r0)2cos2ø), i.e., four
measurements of the mirror at three different angles relative to the
axial supports ranged from 190 to 280 nm.
The optician's calculations and measurements suggest that four
forces with a magnitude of approximately 15 N equally spaced on the
circumference alternating positive and negative will correct the
measured astigmatism. I present calculations using the finite element
method that extend these results.
Calculations
Jim Gunn circulated a CAD drawing (MIRCELL.DXF) on 7/24/98 showing
the 2.5-m primary mirror and the locations of actuators (Figure 1)
proposed to control the astigmatism of the primary mirror surface.
Two rings of 12 actuators at two different radii are proposed. A
similar pattern (green crosses in the Figure) was analyzed because it
was simpler to model; the patterns are so similar that the results
should be almost the same.
A second proposal is to modulate the pressure of a subset of the
existing primary axial support pneumatic pistons. These are arranged
in two
rings. The outer ring contains 30 pistons and the inner ring 18.
The ring radii are approximately the same as those of green crosses.
The model should apply to this proposal also.
The finite element model is described in 2.5-m
primary mirror transverse support system (SDSST Technical
Note 19980713). One-quarter of the mirror was modeled. Symmetric
boundary conditions were applied on the x=0 and y=0 planes. Forces
given by the expression f = 10*cos2ø N were applied to the
nodes corresponding to the green crosses of the Figure. Three cases
were analyzed. Linear superposition should be valid for these results
since the deflections are small and stress stiffening is not
present.
- Only the outer ring of astigmatism actuators are present
(prim94 and prim96).
- Only the inner ring of astigmatism actuators are present
(prim95 and prim97).
- -1 N is applied to the inner ring, +1 N to the outer ring
(prim98).
Figure 1: CAD drawing showing the
2.5-m primary mirror and the locations of actuators suggested by
Gunn (blue circles) These actuators apply forces to correct
astigmatism. Two rings of 12 actuators at two different radii were
suggested. Forces given by the expression 10*cos2ø N were
applied at locations near those of the proposed pattern (green
crosses).
Figure 2: Surface deflection in
meters calculated for astigmatism forces applied to the outer ring
of green crosses shown in Figure 1. (The 9 colors in the legend do
not correspond to the 9 colors in the plot. However, the maximum
and minimum are annotated on the plot and the color order is
green, yellow, red, orange, olive, turquoise, cyan, purple,
magenta).
Figure 3: Residual surface deflection in
meters for Figure 2 with astigmatism removed.
Figure 4: Surface deflection in
meters calculated for astigmatism forces applied to the inner ring
of green crosses shown in Figure 1.
Figure 5: Residual surface
deflection in meters for Figure 4 with astigmatism removed.
The results of the calculations are shown in Figures 2 and 4 for
the two astigmatism cases. The expression uz = a1 +
a2*r2*cos(2ø) +
a3*r2*sin(2ø) was fit to the results and
the residual deflections at each node were calculated. These were
plotted in Figures 3 and 5 for the two cases. The parameters for the
astigmatism fits are given in Tables 1 and 2. A coefficient (as
defined in the Tables) of 143 nm/m2 corresponds to the
Zernike coefficient found by the optician. Consequently, a force
amplitude of 11.0 N and 19.6 N is required for the outer and inner
ring of astigmatism actuators respectively.
Table 1: Surface error fit
parameters for astigmatism forces applied to the outer ring of
Figure 1.
|
prim94
|
uz = a1 +
a2*r2*cos(2ø) +
a3*r2*sin(2ø)
|
|
|
i
|
ai
|
sigma
|
|
1
|
129.9 nm
|
0.161 nm
|
|
2
|
110.9 nm/m2
|
0.126 nm/m2
|
|
3
|
0 nm/m2
|
0.213 nm/m2
|
|
Initial error (RMS)
|
156 nm
|
|
|
Residual error (RMS)
|
5.2 nm
|
|
Table 2: Surface error fit parameters for
astigmatism forces applied to the inner ring of Figure 1.
|
prim95
|
uz = a1 +
a2*r2*cos(2ø) +
a3*r2*sin(2ø)
|
|
|
i
|
ai
|
sigma
|
|
1
|
73 nm
|
0.161 nm
|
|
2
|
62.8 nm/m2
|
0.126 nm/m2
|
|
3
|
-0.2 nm/m2
|
0.213 nm/m2
|
|
Initial error (RMS)
|
88 nm
|
|
|
Residual error (RMS)
|
10.2 nm
|
|
The two support rings of Figure 1 are at approximately the same
radii as the existing primary axial support pneumatic pistons. One
implementation approach consists of replacing some of the pistons in
the outer ring with modified pistons that would allow a force to be
subtracted from the nominal support force. This will reduce the mean
force in the outer ring. Consequently, an analysis of prim98, a case
with -1 N forces applied to the inner ring and +1 N forces applied to
the outer ring, was performed. Prim 98 can be linearly combined with
prim94 or prim95 to understand the effect of subtractive forces on
the mirror figure.
The deformation for prim98 is conical, i.e., linear with radius
(Figure 6). However, it is fit reasonably well by a quadratic (Figure
7). This is equivalent to a change in the radius of curvature of the
mirror but does not introduce significant image aberrations for small
changes. The departure from the quadratic fit is troublesome,
however. Also shown is a r4 fit. The fit is even worse.
However, Figure 8 shows that the deformation can be viewed as a
combination of change of the radius of curvature and spherical
aberration.
Figure 6: Surface deflection in
meters calculated for -1 N applied to the inner ring and +1 N
applied to the outer ring (see green crosses shown in Figure
1).
Figure 7: Quadratic and r4 fit
to the data of Figure 6. The deformation is conical, but the
quadratic fit removes much of the deformation. The r4
fit is not good, but shows the extent to which spherical
aberration can be imposed by means of these forces.
Figure 8: Quadratic and r4 fit
to the data of Figure 6. The combination of r2 and
r4 fits the deformation well.
Conclusions
Two patterns of actuators that apply astigmatism correcting forces
to the 2.5-m primary mirror were examined. The residual error for the
outer ring is less than that of the inner ring, so it is to be
preferred. The residual error for the outer ring is so small that the
addition of the inner ring of actuators appears to be an undesirable
complication. The force amplitude that is required is approximately
11.0 N.
The addition of the inner ring would allow comatic surface error
to be reduced. However, comatic surface error is not readily
distinguished from aberrations due to decollimation of the optics.
Also, the mirror support system is not expected to be a source of
coma. Consequently, it is not likely that the inner ring of actuators
will be useful for this purpose either.
Modulating the force applied by every other or every third
existing axial support piston in the outer ring should be feasible.
This would require modifying 15 or 10 pistons respectively. Further
analysis will indicate if the cos2ø force modulation should be
changed because all actuators are not at the same radius.
It has been proposed that modulating the force applied to the
mirror in the outer ring be accomplished by subtracting astigmatism
correcting forces. This has the effect of reducing the mean force on
the outer ring by 11 N. Scaling the results from prim98, the maximum
departure from a quadratic fit is about 11 nm. The surface error is
6.8 nm RMS. These errors are small enough to be acceptable.
References
1. W.A. Siegmund, E.J. Mannery, J. Radochia and P.E.
Gillett, "Design of the Apache Point Observatory 3.5-m telescope II.
deformation analysis of the primary mirror", Proc. S.P.I.E. 628,
pp.377-389, 1986.
2. "Fabrication of the 2.5 m primary mirror for the Sloan
Digital Sky Survey Telescope, Final report", Optical Sciences Center,
University of Arizona, Nov. 4, 1997.
Date created: 8/5/98
Last modified: 2/5/99
Copyright © 1998, 1999 Walter A. Siegmund
Walter A. Siegmund
siegmund@astro.washington.edu