Error Budget and Wavefront Specifications for Primary and Secondary Mirrors

J. M. Hill

Steward Observatory

Large Binocular Telescope Project

Technical Memo
UA-94-01
August 26, 1994.
http://medusa.as.arizona.edu/lbtwww/tech/ua9401.htm

Abstract

1. Introduction

2 Primary mirrors

3. Secondary mirrors and other optics

4. Mean figure errors

5. Active optics tolerances

6. Near-infrared error budget

7. Conclusions

8 Acknowledgements

9. References

10. Appendix: Other secondaries

Abstract

This memo discusses the allocation of surface errors on both primary and secondary mirrors in the framework of an error budget based on the turbulence spectrum of the atmosphere. The surface error specification changes as a function of the spatial scale on the surface. The entire telescope error budget corresponds to an atmospheric wavefront with r 0 = 45 cm. The primary mirror allocation corresponds to r 0 = 60 cm of which polishing contributes r 0 = 92 cm. Scale factors are introduced to convert from spatial scales on the primary mirror to the spatial scales on the secondary which are appropriate for the same atmospheric turbulence. The secondary mirrors have physical errors similar to the primary mirror while they contribute less to the entire wavefront error of the telescope. Tolerances on mean figure errors such as conic constant and radius of curvature are also discussed.

1. Introduction

The philosophy for the error budget is to match the wavefront produced by the atmosphere in the very best seeing. This should provide images which are limited by atmospheric seeing nearly all of the time. The atmosphere is characterized by Fried's parameter, r 0. The telescope error budget has the goal of meeting a wavefront structure function equivalent to an r 0 = 45 cm atmosphere, or images of roughly 0.23 arcsec FWHM at a wavelength of 0.5 microns. The combined telescope, atmosphere and instrument should deliver a wavefront to the focal plane equivalent to an r 0 = 30 cm atmosphere or a detected image of 0.34 arcsec FWHM. The total telescope error budget has been divided up among its sundry parts according to their relative cost as well as technical difficulty and risk. For example, the axial support of the primary mirrors has been allocated a wavefront distortion equivalent to an r 0 = 214 cm atmosphere. These individual error allocations are then translated into specifications for the various parts of the telescope and its optics. Hill (1990) provides the details of the overall error budget for the telescope.

In the context of this memo, ``specification'' generally means ``an ambitious, but achievable, goal which we will attempt to meet''. The Large Binocular Telescope (LBT, former Columbus Project) error budget has certainly been written in that context. The reader should remember to distinguish between this use of ``specification'' and a contractual specification that might be imposed on an optical fabricator.

The remainder of the introduction summarizes the concept of specifying the optical surfaces in terms of the wavefront structure function. This section was taken from a more elaborate discussion by Hill (1990). Dierickx et. al. (1990) and Dierickx (1992) provide additional details of atmospheric images and their relation to telescope specifications from a diffraction point of view.

1.1 The structure function

Angel (1987) has outlined the error budget strategy for LBT. Since our goal is to build a telescope which degrades the incoming image as little as possible, it seems appropriate to specify the errors in the telescope so they correspond to the distortions already induced by the atmosphere. The wavefront error induced by the atmosphere between two points separated by a distance x is given by

= 0.4175 * ( x / r0 )5/6 waves
where the error is expressed as phase difference . No matter what value of r 0 we use, the errors are always proportional to x5/6 as long as the atmosphere retains a Kolmogorov spectrum of turbulence. This allows us to relax the tolerance on the telescope optics and alignment at large scales since the atmosphere has already distorted the wavefront. We have adopted the ``structure function'' to describe the error in the incoming wavefront as a function of separation. Thus, by selecting a particular value of r 0, we may specify the permissible wavefront distortion induced by a mirror or a telescope. Converting from phase error to a linear dimension, we find the structure function:

where (x) is the root mean square wavefront difference between points on the wavefront with spatial separation x.
( x, , r 0 and should all have the same or consistent units; e.g. meters.)

1.2 Tilt compensation

As we will see in later sections, various parts of the telescope contribute in different ways to the overall wavefront structure function. Tracking jitter introduces tilt of the wavefront or image motion. Alignment errors tend to produce low-order aberrations, while mirror support tends to introduce errors of higher spatial frequency. The large and small scale errors will be subdivided in slightly different ways. For example the primary mirror is not allowed to contribute any tilt to the wavefront since its axis defines where the telescope is pointing. Removing the mean tilt from the wavefront rolls off (reduces) the structure function at large spatial scales as indicated by the correction factor shown here.

D is the telescope aperture diameter.

1.3 Scattering effects

The strict x5/6 power law is not maintained because diffraction effects allow us to relax the specifications at small spatial scales. Ruze (1966) gives the fractional energy loss due to scattering from small errors on scales much less than r 0 as:

where is the rms deviation from the mean wavefront. We want to specify the small scale (1 cm) surface roughness so that no more than 5% of the light is scattered outside the seeing disk at 350 nm. = 12.5 nm implies a 17.6 nm rms wavefront difference or a 9 nm rms surface difference. The overall wavefront error budget can then be specified as the wavefront structure function:

1.4 Zenith angle

When the telescope is looking away from the zenith through more of the atmosphere, we may expect the images to degrade. The seeing degrades according to ( cos z)3/5, so an r 0 = 45 cm atmosphere will be only r 0 = 30 cm at a zenith angle,
z, of 60 ° . We will allow the error budget to relax in this same fashion. Fixing at 12.5 nm, at 0.5 µ m and
r 0 at 45 ( / 0.5 µ m )6/5 ( cos z ) -3/5 cm gives the numerical result of:

2 (x,z) = 3.1*10-16 + 1.65*10-13 x5/3 cos z

for the structure function specification of the telescope error budget. Units have been scaled so that x and are in meters.

Figure 1:This figure shows the wavefront structure functions for an r 0(500nm) = 45 cm atmosphere. The solid line shows the structure function for the Kolmogorov atmosphere. The dot-dash line shows the specification to keep scatter at 6 % for a wavelength of 350 nm. The dotted line shows the atmospheric structure function with tilt removed.

1.5 Telescope error budget

Hill (1990) combined the various parts of the telescope error budget according to the sum of the squares of the image sizes. This works until the error budget gets finely subdivided into a series of structure function specifications. Then, the results need to be combined as the --5/3 power of the adopted r 0. Here we shall modify that treatment by allocating broad categories according to the image size. Finer subdivision will then be done according to the structure function formalism described in the previous section. Because the actual wavefront errors of the telescope do not have a Kolmogorov spectrum, and neither do they produce gaussian profile images, this approach is an approximation at best. However, the rigorous approach only works after you know the functional form of all the errors in the telescope. The error budget for the total telescope is 0.225 arcsec FWHM which corresponds to r 0 = 45 cm. This has been divided as follows:

1.6 Optical surfaces error budget

We expect tolerances for all optical surfaces to be split about equally between design, fabrication, support and thermal control. The primary has a much greater cost, and might therefore expect a larger piece of the error pie for fabrication and support. Smaller optics like the secondaries and tertiaries appear to have a disproportionately small error allocation because the allowable wavefront tolerance scales with the size of the beam. In general, the optical surfaces will be allowed a large fraction of the small scale wavefront error, while the optical design and alignment will use up the large scale error tolerances.

Since optical design has been explicitly removed above, the remainder of this section will be devoted to dividing up the 0.175 arcsec FWHM image size which is allocated to the optical surfaces of the telescope and their supports. This 0.175 arcsec corresponds to a r 0 wavefront distortion of 57.6 cm. (Three significant figures are kept only as an aid to reproducing the calculation later.)

2 Primary mirrors

The primary mirror error allocation is 59.5 cm r 0 wavefront distortion or 0.170 arcsec FWHM image size. These errors include: polishing, testing, coating, axial support, actuator errors, blank fabrication errors, lateral support, wind forces, thermal control and expansion homogeneity. Table 1 shows the error distributed among the categories for zenith pointing and zenith angle 60° . The (cos z)3/5 scaling increases the error budget for the primary to r 0 ~ 45 cm at 60° . Polishing errors are independent of orientation, while lateral support errors are only significant at large zenith angles. An r 0 = 180 cm wavefront corresponds to 0.056 arcsec FWHM or a reflecting surface with errors of 64 x5/6 nm rms, where x is the spatial scale of the errors in meters.

Category 0° 60°
Polishing 123 cm 123 cm
Optical Testing 300 cm 300 cm
Axial Support Distribution 214 cm 150 cm
Lateral Support Distribution --- cm 150 cm
Actuator Errors or Mass Distribution 214 cm 150 cm
Wind Forces on the Mirror 214 cm 122 cm
Ventilation Errors, * T 214 cm 214 cm
Homogeneity Errors, * T 214 cm 214 cm
Reflective Coating 400 cm 400 cm
Primary Mirror Total r 0 -1.67 59.5 cm 45 cm
Table 1: Allowable errors for the fabrication and support of the primary mirror are presented in terms of the r 0 value of the wavefront degradation in centimeters. Errors on scales smaller than 10 cm are not accurately represented by this table. The first column is for zenith pointing and the second column is for zenith angle 60° . The total is from an RSS of the structure functions.

Figure 2: This figure shows the wavefront structure functions for the specification of the primary mirror. The solid line shows the r 0(500nm) = 59.5 cm structure function for the total primary mirror specification. The dotted line shows the r 0(500nm) = 92 cm structure function for polishing alone. The primary has been allowed a 3% loss due to scattering at 350 nm. The dashed line shows the r 0(500nm) = 214 cm structure function for the support of the primary mirror.

2.1 Polishing and testing

For fabrication of the primary mirror surface we have (historically) allocated a r 0 = 123 cm wavefront for polishing and r 0 = 300 cm for testing errors. Because the mirror will be polished on the axial supports, polishing can absorb another r 0 = 214 cm from the support budget provided the axial support pattern does not reverse itself to be worse than r 0 = 150 cm away from the zenith (It doesn't). For spatial scales smaller than 10 cm, where diffraction (scattering) dominates atmospheric wavefront distortion, the surface roughness should be less than 6 nm rms to keep the light loss below 3% at 350 nm.

In reality, the three categories of: ``polishing'', ``optical testing'' and ``axial support distribution'' should be replaced by the longer list of: ``null corrector design'', ``null corrector fabrication'', ``alignment'', ``polishing support'', ``high order surface errors'', ``testing errors'', ``conic constant'' and ``radius of curvature''. These are permitted the total polishing allocation of r 0 = 92 cm (123 cm, 300 cm and 214 cm combined). Later sections of this memo will discuss how the requirements on the low order errors such as ``conic constant'', ``radius of curvature'' and ``astigmatism'' can be relaxed since these errors can be compensated in the telescope. We shall refer to these low-order errors as ``mean figure errors'' since they refer to how well the average surface of the finished optic matches the perfect optical surface if it were perfectly smooth.

The axis of the primary mirror is determined by the optical testing process and is constantly corrected by guiding. We have therefore removed tilt from the wavefront structure function that the figured surfaces are required to match. This substantially tightens the surface tolerances for large scale errors, but makes the tracking and alignment specifications more reasonable.

2.2 Wavefront errors along the way

Since the historical allocations of error in Table 1 were made, the details of many aspects of mirror support, polishing and testing have been worked out. To clarify the error allocations in Table 1, we can ask what wavefront errors we can expect to see in various stages along the way between the polishing machine and installation in the telescope. Hopefully this discussion will provide a clearer distinction between measured wavefront errors and error allowed for in the design process.

Figure Errors are the errors we measure in the mirror surface during the final stages of polishing. If we had an ideal null corrector, repeatable support during testing and no measurement noise, then the measured error in the interferogram would be the actual error on the mirror surface. Given a deterministic polishing process (plus suitable budget and schedule) enough iterations would eventually lead to an error-free mirror. Figure errors are more or less the item referred to as ``polishing'' in Table 1. Thus, we would end the polishing process when the daily interferogram reached a wavefront quality better than r 0 = 123 cm.

Testing Errors are those systematic (fixed in time) wavefront errors produced by the null corrector. Thus, if we polish until achieving a perfect null wavefront, the actual mirror surface will be a reverse image of the imperfections in the null corrector and reference sphere. Axisymmetric errors in the null corrector can be removed from the measured wavefront if they have been calculated from the optical design. Random errors in the null corrector are typically measured and removed by rotating the corrector with respect to the mirror during testing. If the mirror support used during testing is repeatable and if there is no important measurement noise, then the figure at the end of polishing will be the sum of the measured figure errors plus the undetermined residuals of the test optics. This combination has an error allocation of r 0 = 108 cm. The split between testing errors and figuring errors is quite arbitrary, but based on some knowledge of the general performance of null correctors and polishing tools.

Actuator Force Variations can degrade the wavefront if the forces during testing are not repeatable from one day to the next. If the forces in the polishing are repeatable, their values are not very important, since the figuring process will remove any effects of non-uniform support. What is important is that the telescope cell is able to reproduce the same set of forces used during polishing --- whatever they were. (see the LBT Tech Memo on ``Mirror Support System for Large Honeycomb Mirrors II'') The combination of figuring errors, testing errors and actuators force variations has been allocated a combined wavefront error of r 0 = 92 cm. This is the structure function we would hope to meet when the mirror is installed in the telescope mirror cell under the test tower after polishing. Optimization of the support forces via active optics will allow us to recover some of the wavefront error lost to force variations on large scales. So what we really care about are the errors on spatial scales less than about one meter which are difficult to correct with only 100 axial actuators.

At the zenith on the mountain we find additional errors that may arise. These include wind, thermal effects (ventilation and homogeneity) and coating thickness. The combination of these with the above errors seen in the lab brings us to the total primary mirror wavefront budget at the zenith: r 0 = 60 cm.

Away from the zenith we find errors arising from the imperfections of the axial and lateral support patterns. The lateral support deformation of the mirror appears directly as we go toward the horizon. The axial support deformation also appears near the horizon (in reverse) since we polished it out while zenith pointing. The wavefront error allocation has increased to r 0 = 40 cm when 60 degrees from the zenith since the telescope is looking through more atmosphere. This goal is easy to meet since the axial and lateral support deformations are relatively small (Local effects ultimately set the number of actuators that are used.).

3. Secondary mirrors and other optics

3.1 Scaling

Since the incoming light from the primary mirror or entrance pupil is demagnified (reduced in diameter) by the time it reaches the secondary and tertiary mirrors, the optical (physical) tolerances on those mirrors can be correspondingly reduced. An error at 10 cm spatial scales on a 0.8 m secondary affects the 8.0 m primary wavefront at 1 meter scales, and therefore has 10 5/6 more latitude. The optics (all of them) after the primary have been allocated r 0 = 160 cm wavefront error for fabrication and support. The secondary mirror has an error budget similar to the primary in terms of its physical surface. Because the wavefront is demagnified when it hits the secondary, the surface errors scale relative to the atmospheric wavefront so the secondary makes a much smaller contribution to the total telescope wavefront. The geometric scaling factors for some relevant secondaries are given in the following table. These scaling factors are not the same as the secondary magnification.

Secondary Mirror
Identification 		Primary Mirror
Diameter (m)		 Secondary Beam
Diameter (m) 		Scale Factor
MMT/Magellan F/5.2 Cass. 6.502 1.530 4.25
MMT F/9 Cassegrain 6.502 0.966 6.73
MMT/Magellan F/15 Cass. 6.502 0.610 10.66
Magellan F/11 Gregorian 6.502 1.262 5.15
Columbus F/5.2 Cassegrain 8.408 1.838 4.57
Columbus F/15 Cassegrain 8.408 0.719 11.69
LBT F/4 Cassegrain 8.408 ~1 8
LBT F/15 Gregorian 8.408 0.892 9.43

Table 2Secondary mirrors for a number of telescopes are listed here. The rightmost column of this table lists the scaling factor, C, between the beam diameter at the secondary mirror and the primary mirror diameter. This factor is used to adjust the spatial scale of the secondary wavefront when calculating the secondary contribution to the telescope wavefront error. Underfilling of the primary aperture due to infrared configurations with the stop at the secondary has not be considered. The primary mirrors are either 6.502 m F/1.25 or 8.408 m F/1.142. The exact dimensions of the LBT F/4 secondary have yet to be finalized.

Figure3: This figure shows the wavefront structure functions for contributions of the various optics to the overall 8.4 meter telescope. The solid line shows the r 0(500nm) = 45 cm structure function for the total telescope specification. The dashed line shows the r 0(500nm) = 59.5 cm structure function for the total primary mirror specification. The dot-dash line shows the r 0(500nm) = 160 cm structure function for the specification for optics other than the primary. The dotted line shows the r 0(500nm) = 321 cm structure function for the F/15 secondary scaled onto the primary wavefront. Each surface has been allocated a 3% loss due to scattering. Tilt has been removed from the structure functions.

3.1.1Structure function
We may now apply these scaling factors to the spatial coordinate of the structure function to get the appropriate structure function for the physical wavefront (2 x physical surface) of the secondary.

2M2 (x) =( / 2 )2 6.88 (Cx / r0)5/3

where M2 (x) is the root mean square wavefront difference between points on the wavefront with spatial separation (x) on the secondary. C is the ratio of primary diameter to secondary beam diameter.

The astute reader should notice that we have left off the tricky parts of the structure function equation by not including the effects of scattering and removal of the wavefront tilt. The actual equation should be:

2M2 (x) = 2 2 + ( / 2 )2 6.88 ( Cx / r0) 5/3[ 1 - 0.975 (Cx / D)1/3]

D is the telescope aperture diameter. Note that D divided by C is the secondary beam diameter. The equation obviously has problems when x is larger than the secondary beam diameter, which it often is for wide field secondaries.

3.1.2 Scattering
Should the scattering term scale in a different way because the scattering at the secondary occurs in the converging (reimaged) beam? The scattering term is unscaled because it represents light lost into a halo due to high frequency errors on the surface of the secondary. The scattering is from spatial scales much less than r 0. The effective scattering angle (in focal plane units) is different for the secondary, but the amount of light scattered is unchanged. If we want the whole telescope to have less than 6% small angle scattering at 350 nm, then the secondary needs to be as smooth as the primary. Each optical surface needs to keep scattering at the 3% level. (Earlier error budgets allowed only 5% total scatter at 350 nm. It remains to be seen if even this looser specification can be met.) Thus while the secondary seems to make a negligible contribution to the telescope errors on scales of 0.5 meters at the entrance pupil, the secondary contribution becomes much more important on smaller scales.

3.1.3 Tilt removal
The tilt removal term is sufficiently complicated that I'm not completely convinced that the equation above is correct. Here is my best attempt at an explanation so far: For the whole telescope, we are allowed to repoint the primary mirror or tilt the secondary mirror to position the image at an arbitrary location in the focal plane. This effectively removes the net tilt of the Kolmogorov wavefront at whatever speed we make the correction. When discussing the mirror wavefront specifications, we care about the instantaneous image and may therefore ignore net tilt altogether (see discussion in previous section). The secondary structure function gets corrected for the net tilt at the primary mirror rather than for the larger tilt which would occur for a mirror which is the diameter of the beam at the secondary. The alternative argument thinks about scaling r 0 rather than scaling x. The results differ by only about 20% at the largest spatial scales.

3.2 Error budget

Now we can write down a particular (arbitrary) error budget for the secondary mirror wavefront and examine how that structure function scales to that of the telescope as a whole. The secondary error budget is distributed similar to that of the primary except that the error allowances for polishing and testing the convex surface have been increased. Small scale surface errors should be less than 5 nm rms.

ItemPhysical Wavefront r 0
on M2 surface		 Telescope Contribution r 0
scaled for F/5 secondary
Polishing 150 cm 686 cm
Optical Testing 300 cm 1371 cm
Axial Support 280 cm 1280 cm
Lateral Support (280 cm) (1280 cm)
Actuator Errors 280 cm 1280 cm
Wind Forces 280 cm 1280 cm
Ventilation Errors 280 cm 1280 cm
Homogeneity Errors 280 cm 1280 cm
Reflective Coating 450 cm 2056 cm
Total (at zenith) r 0-1.67 74 cm 338 cm

Table 3: Secondary Mirror Error Budget for a particular example --- the Columbus F/5.2 secondary. The second column represents the physical wavefront specification for the secondary expressed in terms of r 0 for an equivalent atmosphere. The third column is the physical specification of the secondary scaled to give its equivalent effect on the wavefront of the telescope at the entrance pupil. The scaling factor is the ratio of the primary diameter to the diameter of the beam at the secondary, which for this example is 4.57. This table does not include scattering effects.

Item Physical Wavefront r 0
on M2 surface		 Telescope Contribution r 0
scaled for F/15 secondary
Polishing 70 cm 660 cm
Optical Testing 100 cm 943 cm
Reflective Coating 280 cm 2640 cm
Axial Support 120 cm 1132 cm
Lateral Support (120 cm) (1132 cm)
Actuator Errors 120 cm 1132 cm
Wind Forces 120 cm 1132 cm
Ventilation Errors 180 cm 1697 cm
Homogeneity Errors 180 cm 1697 cm
Total (at zenith) r0-1.67 34 cm 321 cm

Table 4: Secondary Mirror Error Budget for a particular example --- the LBT F/15 Gregorian secondary. The second column represents the physical wavefront specification for the secondary expressed in terms of r 0 for an equivalent atmosphere. The third column is the physical specification of the secondary scaled to give its equivalent effect on the wavefront of the telescope at the entrance pupil. The scaling factor is the ratio of the primary diameter to the diameter of the beam at the secondary, which for this example is 9.43. This table does not include scattering effects.

3.2.1 LBT wide field optical secondary
The LBT 1.3 m F/4 secondary mirror shares the allocation of error with the wide field corrector. An r 0 = ~ 270 cm wavefront will scale to an equivalent r 0 = ~34 cm physical error on the actual mirror surface. The corrector fabrication will use the remaining r 0 = ~220 cm of wavefront error. In this case, we may be more interested in encircled energy specifications than in the details of the structure function.

3.2.2 LBT F/15 infrared secondary
Since the infrared Gregorian focus has only two reflections and the wavefront is scaled from 8.4 m to 0.89 m, the chopping secondary can probably be fabricated and supported with a r 0 = 321 cm wavefront (physically equivalent to r 0 = 34 cm). The remaining 200 cm error can be contributed to the chopping motion which needs all the help it can get. The discussion of the near-infrared error budget (see below) may suggest that the F/15 secondary should have an even higher quality physical surface. But, r 0 = 34 cm is still being proposed as the spec.

The accompanying plot (Fig 4) shows the specs for secondary figuring in terms of physical wavefront on the mirror surface.

3.2.3 Other secondary mirrors
Some additional secondary mirrors are tabulated in the appendix.

Figure 4: This figure shows the wavefront structure functions for polishing allocations for the various optics. These are the structure functions at the physical surface of each mirror. The solid line shows the r 0(500nm) = 92 cm structure function for the primary polishing specification. The dotted line shows the r 0(500nm) = 34 cm structure function for the F/15 secondary polishing specification. The dashed line shows the r 0(500nm) = 34 cm structure function for the F/4 secondary polishing specification. Each of these structure functions was calculated with a 3% loss due to scattering.

4. Mean figure errors

This section discusses the fixed low-order errors which may be left in the primary and secondary mirrors. Because these errors can be compensated in the telescope through alignment or force changes, they are allowed larger deviations from the nominal error budget.

4.1 Assumptions

Assume that the geometrical spotsize tolerance for mean figure errors is 0.024 arcsec rms image radius (weird units calculated by my third order design program). This corresponds roughly to 0.056 arcsec FWHM images or 180 cm r 0. This spotsize is equivalent to 0.73 microns of wavefront spherical aberration on an 8.4 m telescope, which exceeds the structure function for a 180 cm r 0 atmosphere. For the discussion below, this is the amount of spherical aberration we will allow and still consider the telescope to have an acceptable image. Obviously this isn't a hard number and has to be balanced against other parts of the telescope and polishing error budgets. Also note that this term does not appear explicitly in the error budgets discussed above, except as it is included in the polishing error term.

In Figure 5, the structure function for 0.73 microns of spherical aberration in the wavefront is plotted. On large scales, the error is similar to the r 0 = 92 cm polishing specification while the spherical aberration contribution is almost negligible on spatial scales smaller than 10 cm. I guess the lesson here is that proper spacings of the mirrors and test optics is important --- a truth known even to the general public in the post-Hubble world.

Figure 5: This figure shows the wavefront structure functions for polishing the primary compared to the amount of spherical aberration we have allowed for mean figure errors. The solid line shows the r 0(500nm) = 60 cm structure function for the primary mirror specification. The dotted line shows the r 0 = 92 cm structure function for the primary polishing specification. The dashed line shows the structure function for 0.73 microns of wavefront spherical aberration.

These issues apply to all the large telescope projects. I have chosen to examine the LBT primaries and secondaries in detail, but all the mirrors have similar amplitudes. Other mirrors are listed in Table 5.

4.2 Asphere tolerances

4.2.1 Primary asphere
LBT 8.4 meter For the LBT 8.408 m F/1.142 primaries, if the focal plane is fixed the primary asphere can vary over the range --0.99993 to --1.00007 (± 6.7 * 10-5) and still produce acceptable images (< 0.024 arcsec rms image radius).

MMT/Magellan 6.5 meter For the MMT / Magellan 6.502 m F/1.25 primaries the primary asphere can vary over the range --0.99991 to --1.00009 (± 8.7 * 10-5).

Too tight? This is the most simplistic tolerance for the primary asphere and the hardest one to meet (maybe impossible). To relax the tolerance on the primary mirror, we need to allow the back focal distance to change after the mirror is installed in the telescope. The primary mirror support allows a millimeter or two of motion, but that isn't enough to help. (see below)

4.2.2 Secondary asphere
LBT F/15 For the LBT F/15 focus, if the focal plane is fixed the secondary asphere can vary over the range --0.7321 to --0.7331 (± 5.0 * 10-4, ± 680 ppm) and still produce acceptable images (< 0.024 arcsec rms image radius).

LBT F/4 For the LBT F/4 focus, if the focal plane is fixed the secondary asphere can vary over the range --3.2347 to --3.2377 (± 1.5 * 10-3, ± 460 ppm) and still produce acceptable images (< 0.024 arcsec rms image radius).

4.2.3 Refocus
LBT F/15 For the LBT F/15 focus, based on the spherical aberration you get from refocussing the telescope, 0.024 arcsec rms radius allows the focal plane to move by±27 mm. ("refocus" is used to mean moving the secondary to reposition the focal plane.)

LBT F/4 For the LBT F/4 focus, based on the spherical aberration you get from refocussing the telescope, 0.024 arcsec rms radius allows the focal plane to move by±2.1 mm.

4.2.4 Move focal plane
We can take advantage of the spherical aberration generated by refocussing the secondary and moving the focal plane to compensate errors in the primary or secondary aspheres.

To compensate the primary asphere If we allow the focal plane to be mechanically moved by ±20 mm relative to the primary vertex, then we can accommodate a primary asphere between --0.99937 and --1.00063 ( ± 6.3 * 10-4) with the LBT F/4 focus, and between --0.99995 to --1.00005 ( ±5.0 * 10-5) for the LBT F/15 focus.

This is good news and bad news. The good news is that the F/15 focus can accommodate a larger range of primary asphere, so shims are less likely to be needed for repositioning the instruments. That bad news is that if the primary is wrong by more than 0.0002 in the asphere, it will be extremely difficult to correct the problem by adding or removing shims. The problem is how to implement shims more than a few centimeters thick on the instrument rotator that must be extremely stiff to support the instrument. Alternately, we could make adjustments to the instruments, but that means that they all have to be redesigned after first light. The MMT Conversion has an even tougher problem, because it has used up all of the space between the floor and the mirror cell.

I do not yet have a detailed plan for how to build this spacing change into the telescope, but it appears to be necessary. This probably means that we will have to design shims into the mount for the instrument rotator bearing. Mirror cell designers take note!

We also note that different Cassegrain focal ratios require varying amounts of focal plane motion to compensate an error in primary asphere. For example, moving the Columbus F/5.17 focal plane from 2.00 m to 2.01 m corrects for a primary asphere of --1.000184, but the same focal plane motion on the F/15 focus corrects for a primary asphere of --1.000024. Thus an error in primary asphere causes the coincident focal planes to separate, so this restricts some of our mechanical options.

To compensate the secondary asphere Alternately, if we allow the focal plane to be mechanically moved by ±20 mm relative to the primary vertex, then we can accommodate a secondary asphere between --3.222 and --3.250 ( ±1.4 * 10-2, ± 4200 ppm) with the LBT F/4 focus, and between --0.73221 to --0.73297 ( ±3.8 * 10-4, ± 520 ppm) for the LBT F/15 focus.

4.2.5 Conclusions on aspheres
The simple conclusion is that the primary asphere needs to be correct (or correctable) to a few parts in 10000. Jim Burge's 1991 null lens designs showed a RSS tolerance error in the conic constant of 0.00015 and a worst case error of 0.00037 assuming that the optician and mechanical designer both work very hard. See Burge (1992a,b) for a description of the null correctors. The initial tolerances on the pentaprism test suggested that we could detect primary conic constant errors to 0.0001. Note that the pentaprism test has now been replaced with a holographic null. Burge et. al. (1994) report that both the null corrector and the computer generated hologram can measure the 6.5 meter F/1.25 primary parabolas to 0.0001.

Focal plane adjustment of ± 20 mm looks like a prudent option for LBT F/4, but that adjustment probably is not useful for F/15.

The secondary asphere for F/15 should also be correct to a few parts in 10000. That also appears to be practical. The secondary asphere for F/4 appears more tolerant, but that needs more analysis which includes the corrector. Burge reports that the F/4 conic can be measured to 1 part in 10000 (100 parts per million).

4.3 Focal length tolerances

Analyzing focal length tolerances is somewhat trickier than asphere tolerances since the telescope always needs to be refocused before evaluating the change in image quality.

4.3.1 Primary focal length
LBT F/4 For the naked F/4 focus of LBT, increasing the primary focal length by 10 mm (radius of curvature by 20 mm), is equivalent to changing the primary asphere from --1.000000 to --1.000187 if we also decrease the back focal distance by 16.5 mm. If the back focal distance remains fixed, the primary focal length may change by 2.3 mm before exceeding the 0.024 arcsec rms image radius (with a small change in system focal ratio from 4.000 to 4.002).

LBT F/15 For the LBT F/15 Gregorian focus, increasing the primary focal length by 10 mm, is equivalent to changing the primary asphere from --1.000000 to --1.000032, if we also decrease the back focal distance by 22.7 mm. Thus, changing the back focal distance from --2.500 m to --2.478 m and changing the primary asphere from --1.000000 to --0.999968 will recover the performance lost when the focal length increased by 10 mm. If the primary asphere is not adjusted, then the change in primary focal length is limited to 21.2 mm, with either a corresponding 48.2 mm decrease in back focal distance, or a 3.8 mm increase in secondary focal length, before exceeding the 0.024 arcsec rms image radius (at fixed system focal ratio). If the F/15 back focal distance, secondary focal length and primary asphere remain fixed, the primary focal length may change by 27 mm before exceeding the 0.024 arcsec rms image radius (with a small change in system focal ratio from 14.707 to 14.781). See the section below on combined tolerances.

Combined parameters If either, the back focal distance and final focal ratio, or the back focal distance and primary asphere, are adjusted together; very large changes (10%) in the primary focal length can be accommodated without degrading the images. (Byard and Bonaccini have verified the third order calculations by ray-tracing with CodeV while varying the primary radius by ± 150 mm.) The problem with these large changes is that they result in a whole new telescope design.

Constraint of the null corrector The primary radius seems to be a less sensitive parameter for the single Cassegrain focus. But, the primary radius is also important because it must match the null lens design. (Jim Burge has included a 1 mm radius error in the null lens tolerances.) The null lens sets a substantially tighter tolerance on the radius than the telescope does. To meet the asphere tolerances discussed above, the LBT primary focal length must match the null corrector (or vice versa) to 0.6 mm (or the radius must match to 1.2 mm). (If the conic error produced by the null corrector when the primary radius is wrong the same as the conic compensation I discussed in the previous paragraph, then we might be able to make a good telescope simply by changing the back focal distance. That would relax the radius requirement on the primary from the null corrector. But the effects probably don't cancel since the null corrector and the secondary are at different conjugates of the primary. This same discussion also applies to the secondary test optics.)

Constraint of interferometry For the single primary mirror telescopes, the radius of curvature should not be a problem. The problem for LBT when working in two-shooter mode is that the focal lengths of both primaries need to be matched at about the 0.25 mm level. This is to match the full field of diffraction images in interferometric mode. See the discussions in Hill (1990) and Hill (1994).My current thinking is that the interferometric mode will not tighten the radius tolerance (beyond 1 mm) because we will be able to compensate primary/secondary radius errors by adjusting the camera and collimator optics in the reimaging stage.

4.3.2 Secondary focal length
LBT F/4 For the naked F/4 focus of LBT, increasing the secondary focal length by 10 mm (radius of curvature by 20 mm) and refocussing increases the back focal distance by 32 mm. In this case, both the primary and secondary aspheres remain the same and the image quality is preserved over large ranges in back focal distance. If the focal plane remains fixed, then the secondary focal length can only change by 0.7 mm before exceeding the 0.024 arcsec rms image radius.

LBT F/15 For the LBT F/15 Gregorian focus, increasing the secondary focal length by 10 mm and refocussing increases the back focal distance by 128 mm. If the focal plane remains fixed, then the secondary focal length can only change by 2.1 mm before exceeding the 0.024 arcsec rms image radius.

As with the primary, the secondary focal length is not restricted by telescope image quality, but rather by how much we are able to shift the telescope focal plane. The wide field corrector and secondary test optics will also place limits on the secondary focal length. The secondary asphere and radius will be checked with a swing arm profilometer during fabrication.

4.3.3 Conclusions on radii
The primary radius of curvature appears to be completely limited by the null corrector to a tolerance of about 1 mm (focal length tolerance of 0.5 mm). However, this should also be checked when tolerancing the F/4 corrector and secondary.

The LBT F/15 secondary radius of curvature needs to be correct to about 4 mm (focal length tolerance of 2 mm), but see the caveats about the combined focus.

4.4 Combined tolerances

Primary focal length and asphere Dan Blanco has pointed out that the radius and asphere parameters can be related in a way which gives zero focus shift for the zero spherical aberration condition. Several examples of this effect can be seen above. At the Columbus F/5.17 focus, a 10 mm change in primary focal length can be compensated by changing the asphere to --1.000024, while still using the same secondary and not moving the focal plane. What limits this relation? The further from nominal RC condition the mirror gets the more coma develops. Astigmatism is also changing, but more weakly. There are two problems with this approach. First, the compensation factor is a function of secondary magnification. Second, the errors produced by the null optics don't allow us the luxury of independent control of the radius, since the radius is measured independently from the null wavefront. This compensation effect will be interesting when we try to control the asphere by bending the mirror with actuators.

Secondary focal length and back focal distance For the example of secondary focal length for LBT F/15 in the previous section,

| L2 - BFD / 12.8 | < 2.1 mm

and for LBT F/4,

| L2 - BFD / 3.2 | < 0.7 mm

The range of secondary focal length, L2, and back focus, BFD, are limited only by the acceptable final focal ratio. These same relationships can be calculated for the other telescopes using the values of BFD and L2 from Table 5.

4.5 Wide field corrector

The remaining constraint on the primary/secondary asphere/radius comes from the wavefront that can be corrected by the wide field corrector. This is unlikely to be a problem, but we should check. If there is a problem, it can be completely avoided by not figuring/assembling the corrector until after the telescope has first light.

Dan Fabricant in a February 1993 memo describes work done with Harland Epps to evaluate the sensitivity of the MMT Conversion F/5 corrector to the primary conic and focal length. They find that if the primary conic is held to ± 1.0 * 10-4, and if the focal length is held to ± 1.5 mm, then acceptable images can be obtained through the corrector by refocussing the telescope. Presumably slightly larger variations than this can be tolerated if we are allowed to shift the corrector and respace the elements. Large variations in the primary conic (± 5.0 * 10-3) are definitely not acceptable without completely redesigning the corrector.

4.6 In-telescope testing

One alternate strategy is to fabricate the secondaries after you have measured the primary asphere in the telescope or with the pentaprism test. This allows the secondary radius and asphere to compensate for the primary without moving the focal plane. Domenico Bonaccini has looked at this problem with CodeV and found that the primary asphere can be allowed to vary significantly. For example, if the primary asphere came out to be --1.001, we could make the RC version of the F/15 secondary and still have a good F/5 focus. (This discussion has occurred before, circa 1990.) This might suggest a strategy where we tried to make the primary asphere just slightly hyperbolic (~--1.0003) so that compensation by refiguring the secondary is easier than with an ellipsoid. The obvious drawback to this plan is that you can't finish the secondaries until you have finished the primaries. You also need a very accurate way to test the primary in the telescope. If the primary asphere gets more than a few waves away from a parabola, the testing at prime focus becomes more difficult. I've assumed that the primary would be tested at prime focus in the telescope, since if you could test it in the optical shop, then you could fix the conic constant during final figuring. Conversely, if you have a good shop check of the asphere (in addition to the null lens), there is no need to bother with the prime focus tests. The general plan we have adopted is to make both primaries and secondaries in parallel. In-telescope testing is then reserved as somewhat of a safety margin if problems are encountered.

Focus 1
conic		 L1
mm		 BFD
mm 		2
conic 		L2
mm
LBT F/4 6.7E-5 2.3 2.1 1.5E-3 0.7
LBT F/15 6.7E-5 27 27 5.0E-4 2.1
Columbus F/15 6.7E-5 27 27 9.3E-4 2.1
MMT F/15 8.7E-5 25 25 1.1E-3 2.1
MMT F/9 8.7E-5 9.6 9.4 9.2E-4 1.4
MMT F/5 8.7E-5 3.5 3.3 8.4E-4 0.8
Magellan F/11 8.7E-5 14 14 3.3E-4 1.6

Table 5:Tolerances on conic constants and focal lengths for a selection of telescope focal stations. These tolerances are for image radius less than 0.024 arcsecond rms as calculated from third-order parameters. Compensation by other telescope parameters has not been included. The primary asphere may vary by ± _1 around the nominal parabolic value of --1.000. The primary focal length may vary by ± L1 mm around the nominal value (allowing the telescope focal ratio to change), but the primary focal length is more tightly constrained by the null corrector than by the telescope. The back focal distance (below the primary vertex) may vary by ± BFD mm from the nominal value. 2 and L2 are the corresponding secondary tolerances.

5. Active optics tolerances

Another question to address is: How much focus, spherical aberration and astigmatism will we permit to be subtracted off the finished mirror by adjusting the mirror support actuators (active optics)? The largest limit is set by how much bending force the actuators can safely apply to the mirror. A more practical limit is set by how much spherical aberration or astigmatism we can introduce without introducing higher order errors.

This discussion comes primarily from a November 1991 memo by Byard and Bonaccini. We consider now the possibility of correcting an error in the conic constant, or k, by the application of appropriate forces to the actuators. The equation for the sag, Z, of a general conic is:

where c is the curvature 1/R and h is the radial position on the mirror. Differentiation shows that an error d in the conic constant results in a difference in sag which is given (for the case of a parabola) by:

dZ =( h4) / (8R3) * d

where R is the radius of curvature.

The departure of one surface from another which follows a fourth order power law can be described by a combination of the first two purely radial Zernike terms,

Zr1 = 2 2 - 1

and

Zr2 = 64 - 62 + 1

where

= h / R

So we can write

or

This means that a linear combination of only the first two purely radial terms with the above ratio of coefficients is needed to correct errors in the conic constant.

The actuator forces required to correct primary mirror aberrations described by various low-order Zernike polynomials have been calculated by Parodi using Finite Element Analysis in BCV Reports 136 and 140. His results show that to correct 1000 nm errors introduced in the form of the first three radial Zernike polynomials requires maximum actuator forces of around
± 300 N for Zr1 (focus) and ± 2400 N for Zr2 and ± 5400 N for Zr3. In some cases these forces are applied to two adjacent actuators to produce a bending moment near the edge of the mirror. If we assume that the maximum force to be applied to any one actuator should not exceed 600 N or approximately 60% of the forces to to support the mirror against gravity, then the maximum values the coefficients for the first three Zernike coefficients can assume are 2000, 250, and 110 nm respectively. The 600 N force is approximately the mean force required to support the mirror against the gravity load or 50% of the peak axial force when zenith pointing.

With a horizontal primary, the maximum sag at the mirror edge ( = 1) that can be corrected with the requirement that the force on any one actuator does not exceed 600 N is, by inspection of the data provided by BCV, 1000 nm. This means that the maximum primary conic, d, which is correctable with the support actuators is ± 0.0002.

If we allow additional forces to be applied to the edge of the mirror, the BCV results indicate that the peak correction forces can be reduced by a factor of 4 to 10. This would allow a substantially larger conic correction. But, at that point we must worry about residual figure small scale errors caused by bending the mirror.

6. Near-infrared error budget

The previous discussion has only considered the performance of the telescope and the atmosphere at a wavelength of 0.5 µ m. For an r 0 of 45 cm at 0.5 µ m we will expect a 0.23 arcsec FWHM image. With an 8.4 meter telescope, this means the telescope diameter is nearly 20 times r 0. From Roddier (1990) we see than correction of tilt alone (rapid guiding) can improve the image by no more than about 20%. Because r 0 increases with wavelength as 1.2, at 2.0 µ m r 0 has increased by 4 1.2 to 237 cm. This produces a 2.0 µ m image of 0.17 arcsec FWHM with D/r 0 now equal to 3.5. Rapid guiding in the near infrared under these conditions can improve the image quality by a factor of as much as 5 to produce a near diffraction-limited image. The main point here is that the telescope will produce diffraction-limited images at 2 µ m if the error budget is able to stay within limits of the r 0 = 45 cm Kolmogorov atmosphere with tilt removed. Because we are near the diffraction-limit, the infrared error budget should probably be tabulated in Strehl ratio rather than image quality.

7. Conclusions

It appears that the planned optical fabrication techniques are able to meet the desired telescope tolerances.

8 Acknowledgements

I would like to thank Dan Blanco and Buddy Martin for the many hours of interesting and helpful discussions we have had on this topic over the last five years Thanks also to Domenico Bonaccini, Jim Burge, Paul Byard, Warren Davison and Steve West for their contributions to these calculations.

9. References

10. Appendix: Other secondaries

Item Physical Wavefront r 0
on M2 surface 		Telescope Contribution r 0
scaled for F/9 secondary
Polishing70 cm 471 cm
Optical Testing 100 cm 673 cm
Reflective Coating 280 cm 1884 cm
Axial Support 120 cm 808 cm
Lateral Support (120 cm) (808 cm)
Actuator Errors 120 cm 808 cm
Wind Forces 120 cm 808 cm
Ventilation Errors 180 cm 1211 cm
Homogeneity Errors 180 cm 1211 cm
Total (at zenith) r0-1.67 34 cm 231 cm

Table 6: Secondary Mirror Error Budget for a particular example --- the MMT F/9 Cassegrain secondary. The second column represents the physical wavefront specification for the secondary expressed in terms of r 0 for an equivalent atmosphere. The third column is the physical specification of the secondary scaled to give its equivalent effect on the wavefront of the telescope at the entrance pupil. The scaling factor is the ratio of the primary diameter to the diameter of the beam at the secondary, which for this example is 6.73. This table does not include scattering effects.

Item Physical Wavefront r 0
on M2 surface 		Telescope Contribution r 0
scaled for F/11 secondary
Polishing 70 cm 361 cm
Optical Testing 100 cm 515 cm
Reflective Coating 280 cm 1442 cm
Axial Support 120 cm 618 cm
Lateral Support (120 cm) (618 cm)
Actuator Errors 120 cm 618 cm
Wind Forces 120 cm 618 cm
Ventilation Errors 180 cm 927 cm
Homogeneity Errors 180 cm 927 cm
Total (at zenith) r0-1.67 34 cm 175 cm

Table 7: Secondary Mirror Error Budget for a particular example --- the Magellan F/11 Gregorian secondary. The second column represents the physical wavefront specification for the secondary expressed in terms of r 0 for an equivalent atmosphere. The third column is the physical specification of the secondary scaled to give its equivalent effect on the wavefront of the telescope at the entrance pupil. The scaling factor is the ratio of the primary diameter to the diameter of the beam at the secondary, which for this example is 5.15. This table does not include scattering effects.