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6.2 Special considerations for astronomical telescopes.

Astronomical telescopes form a class of systems which are ideally suited to measurement with the curvature technique using the ef program. In this case we can benefit from the ready availability of broad band plane wavefronts (stellar sources), and from the optical smoothing due to atmospheric seeing. Assuming a typical seeing FWHM of 0.5", then to first approximation the number of pixels required across an astronomical extra-focal image to give critical sampling is given by:
n > 2*l/(2.5e-6 f*f*D).
Where l is the extra-focal distance, f the system f-ratio and l the pupil diameter. For a 4m telescope at f/35 and a 1m extra-focal distance, this indicates an image diameter of greater than 200 pixels.

6.2.1 Atmospheric seeing.

The extra-focal image technique is relatively insensitive to seeing provided exposures are long enough to average the atmospheric wavefront. The presence of atmospheric seeing makes it mandatory to obtain fairly long exposures for each extra-focal image. We generally recommend at least 30 seconds be used. The presence of seeing will slightly reduce the magnitude of the curvature signal, and thus cause the program to systematically underestimate the telescope aberrations. The worse the seeing, the more the telescope aberrations will be under estimated. Under typical conditions, the under-estimation is less than 10and can be safely ignored. The actual magnitude of the effect is mainly dependent on the the size of the telescope, Frieds parameter and the spatial frequency of the aberration to be measured.

More seriously, dome seeing can introduce aberrations which persist for several 10's of seconds. This can lead to jitter in the reconstructed aberration values. The long time-scale aberrations are low spatial frequency mainly tip-tilt focus and astigmatism. A good diagnostic for this effect is to see if the reconstructed wavefront jitter is dependent on zenith angle.

Residual error in wavefront determination as a function of the number of co-added frames. Frames are actually separated by 0.1 seconds assuming a 20m/second wind speed, and Frieds parameter of 30cm (at 0.55 microns) on a 4m telescope. These results are derived from a diffraction simulation of the imaging process. The dotted line represents the residual error that would be expected from averaging a number of independent wavefronts.
 

In the above figure we show the results of reconstructing a wavefront from a series of simulated extra-focal images. The simulation is a full Fresnel diffraction calculation, using a Kolmogorov phase screen to perturb the wavefront. The errors induced in the reconstructed wavefront appear to be 2 to 3 times worse than would be expected if we were just looking at residual wavefront error. The increase in error is probably due to aliasing. However the errors all fall as would be expected with increasing numbers of frames. The focus error is somewhat correlated between frames so doesn't drop quite as fast as the others. With real measurements you might also expect to see higher order aberrations fall to zero slightly faster than shown in these graphs. You can use these graphs to estimate the error due to seeing on your reconstructed wavefronts as long as you have some idea of Frieds parameter and the effective wind speed.

Laplacian Optics Inc.     Email:   laplace@laplacian.com

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