CURVATURE SENSING FOR ALIGNMENT OF HET AND SALT

INTRODUCTION

The Hobby-Eberly Telescope and its southern hemisphere counterpart, the Southern African Large Telescope, have fairly tight tolerances on the alignment of their 91 primary mirror segments, in both the tip/tilt and global radius of curvature degrees of freedom. The tolerances are not identical on the two telescopes, but in round numbers they are about ±0.1 arcseconds (one dimensional rms on the sky) in tip/tilt, and ±10 microns on the radius of curvature.

In both telescopes this alignment is accomplished using a fixed reference source at the center of curvature (with a shearing interferometer for HET and a Shack-Hartmann array for SALT). In this note, I consider the alternative technique of curvature sensing. This would have the advantage of being done on-sky, which should save time compared to moving the telescope to point to the center of curvature. In addition, one could map out azimuth-dependent effects, and there may be less vulnerability to various other systematic effects than with the present fixed reference approach. Note that curvature sensing, like the present approaches, is a geometrical optics test, and therefore is not sensitive to segment piston errors, which we do not consider in this note.

In curvature sensing, a difference image is formed by subtracting inside-of-focus and (appropriately rotated) outside-of-focus images. The intensity of the difference image is then proportional to the Laplacian of the wavefront phase. Since many common aberrations (including tip/tilt and astigmatism) have vanishing Laplacian, this latter information must be supplemented by boundary conditions (also contained within the difference image), which define the normal derivative of the wavefront phase. A significant complication for segmented mirrors is that the boundary conditions appropriate to segment aberrations from different segments overlap, so that the resulting equations are coupled. However, we show below that, because we are interested only in the restricted problem of segment tip/tilt plus global radius of curvature, the extraction of the desired information from the difference image is still relatively straightforward. Another shortcoming of curvature sensing - that it requires very bright stars because the images are substantially out of focus - is a concern but does not present insurmountable difficulties.

In the following I first estimate the sensitivity of a curvature based method to the HET/SALT alignment parameters of interest. I then present some preliminary Monte Carlo calculations which illustrate the proposed method in more detail.

ESTIMATED SENSITIVITY

The idea of the restricted curvature sensing problem is simple enough: tip/tilt errors cause the segment contribution to the image to be translated in the out of focus image; radius of curvature errors cause the contribution to be expanded or contracted. In either case one essentially looks for the edge of the segment contribution and measures its displacement from its ideal position. Since everything is defined by simple geometrical optics, it is easy to estimate the size of the relevant effects:

1. For tip-tilt errors (measured on the sky):

A rough guess for the sensitivity comes from taking α = 1 pixel; a few tenths of a pixel would represent a more difficult measurement, but should not be impossible. Using the above value for α and assuming F = 4.3 and p = 24 microns, I find δθ = 0.12 arcseconds on the sky (or 0.06 arcseconds at the segment), which is at least consistent with the desired sensitivity. These measurements are modestly overconstrained - 306 edges (both inner and outer edges are useful) to determine 182 tip/tilt angles - so one can expect additional modest improvement (perhaps 30%) over this estimate.

2. For radius of curvature errors:

Note that if there are re-imaging optics, both F and δf refer to the final telescope values, not those corresponding to the primary mirror. Denoting the latter quantities with a subscript 1, we have:

Finally, δf₁ can be related to the radius of curvature error δr by δr = 2 δf₁ so that

Taking β = 1, n = 11, F₁ = 1.3, and the other numbers as above, I find δr = 208 microns. In this case, the measurement is highly overdetermined, since there are 306 measurements to determine this single parameter. I estimate that this would reduce the final error by √306 = 17.5 to 12 microns, which is again close to the required value. I note that visual inspection of preliminary out-of-focus images from HET show the characteristic "web" structure associated with this effect; this provides additional evidence that the sensitivity of curvature sensing to this parameter lies in a useful range. [Note that a radius of curvature error is equivalent to (identical) focus errors on each individual segment. Since focus has a non-vanishing Laplacian, there is also an intensity effect in addition to the edge effect associated with this parameter, but I do not consider that additional information in this note.]

MONTE CARLO SIMULATIONS

In these simulations, I generated difference images for both standard and random misalignments by straightforward geometrical optics techniques. The signature of the standard misalignments was determined by summing the pixels in a small box centered on each intersegment edge in the difference image. The random misalignments were then analyzed by measuring the corresponding boxes on their difference images, and the problem was inverted via Singular Value Decomposition. Optical parameters appropriate to the HET were used: 24 micron pixels and an overall focal length of 43.04 meters. I performed four different simulations: without and with seeing (1 arcsec in the long exposure limit) for an overall out-of-focus distance corresponding to a 60 arcsecond image and to a 20 arcsecond image. These out-of- focus distances reduce the effective brightness of the star by about 9 and 6.5 magnitudes, respectively. The simulations without seeing are not intended to be realistic, but are meant to provide a reference against which the degradation associated with seeing effects can be measured. In addition to the effects of seeing in the long exposure limit, these simulations include the effects of pixellation of the images, of cross-talk between tip/tilt and radius of curvature, and of the "nearest neighbor approximation." [In the presence of seeing, the boundary of one segment image extends in principle beyond its nearest neighbor segments, an effect which we ignore here.] The simulations do not include the effects of read noise in the CCD, of counting statistics, of incomplete averaging of seeing effects, or of higher order segment aberrations. In addition, I assume that the location of the optic axis of the system is perfectly known in advance, as is the overall out-of-focus image size, and the orientation of the detector relative to the sky; a more realistic algorithm would have to determine these parameters and would do so with some associated uncertainty, which in turn would contribute to the overall errors. There will be a loss of three degrees of freedom associated with uncertainties of this latter type: global tip/tilt and global rotation or curl. The loss of global tip/tilt is inconsequential because it simply amounts to having to repoint the telescope by a very small amount; indeed, it is preferable to repoint the telescope, rather than using up the actuator dynamic range for this purpose. However, some attention does ultimately need to be paid to the loss of global rotation sensitivity, but we do not consider this further here.

The results of the simulations are summarized in the following table.

Before and after difference images are shown for Trials 2 and 4 (the trials with seeing) in Figures 1 and 2 below. [Not all of the 1024 x 1024 pixels on the detector are shown.] We stress that the "After" columns in the above Table should not be interpreted as the ultimate uncertainties associated with the method, as not all sources of error have been included in the simulations. Nevertheless, the results are encouraging. These simulations suggest that for large images, good convergence can be obtained for both tip/tilt and radius of curvature, and good radius of curvature measurements can be made even with relatively small images. At this point, rather than pursue more sophisticated simulations, it may be easier and more reliable simply to try the method out at the HET.

It is possible that this method could be adapted to the proposed extremely large telescopes of the future, such as the California Extremely Large Telescope (CELT). For constant segment size, a figure of merit for the method is the atmospheric coherence diameter divided by the number of segments across the diameter of the telescope. CELT has three times the number of segments across the diameter, but the value of r₀ may be twice as large, so that the figure of merit may be only about 30% less than that for HET.