Target Acquisition with NIRSPEC

Centering of Targets on MEMS Slit

December 11, 2002

The Problem

The strategy for target acquisition with NIRSPEC is to image the field with all facets open, then measure the position of a pre-selected star and command the telescope to perform a small angle offset. For this to work, the position of the star has to be measured with an accuracy of about 10% the size of a pixel, i.e. 10 mas. Here we investigate how difficult it is to achieve this accuracy given the degradation of the PSF by the MEMS and NIRSPEC optics.

The Approach

The issue was addressed by simulating the MEMS degraded PSF using

All other assumption on facets, oversizing, pixel sizes are the same as before. The PSF on the detector was "observed" with 100mas pixels. This was done assuming several different relative offsets between MEMS facets and detector pixels. Two examples of the geometry are shown in figure 1. The green grid with black gaps are the facets. Superimposed is the the pixel grid of the detector as white lines. The right panel shows the case where the facets and detector pixels are aligned. The left part shows the case where the is an offset of 50mas in x-direction between facets and pixels, but the alignment in y-direction is still perfect. Computation were also carried with offsets of up to 50mas in y-direction. figure 1.
For each configuration, the PSF was placed at different places on the MEMS facets. The position of the target on the observed image was estimated from its center of gravity. For each position, the center of gravity was computed as
cog=Σ[x * f(x)] / Σ f(x)
where x is the position on the detector and f(x) is the detected image. The detected cog was then compared with the true cog computed from the PSF before it enters the spectrograph.

Results

λ=2.5μ

Figure 2 shows the resulting difference between the true and measured cog as a function of the x position of the PSF. The x position is given in units of facet coordinate. Zero means that the star is located between two facets, whereas a value of x=0.5 means the star is located in the center of a facet. Calculations for different y-positions of the PSF and different offsets between facets and pixels are shown.

The color of the curve in the upper panel indicates the y-position of the PSF relative to the facets. The black curves correspond to cases when the PSF passes through the center of the facets as it is moves along the x-direction. The green curves are the cases where the PSF is centered on a gap. The red curve is the intermediate case. The line style of the curves mark the different relative offsets between facets and pixels as shown in figure 1. The solid curve corresponds to a perfect registration of facets and pixels as shown in the right panel of figure 1. The dotted curves correspond to the left panel where there is a mismatch between pixels and facets. The dashed curves are the intermediate cases.

figure 2

Using the cog as estimated target position works well for a noiseless simulation. However, application to noisy data would require to restrict the number of pixels used to compute the cog to a small region around the brightest pixel. This is true for any algorithm to estimate target positions. I therefore have also computed the cog on the observed image using only a 3x3 pixel region around the brightest pixel. This is shown in the lower panel of figure 2. The green lines are the same as in the upper panel. The blue lines show the same cases but with the cog computed from the smaller region. As expected, the differences between the true and the measured position increases when the region is restricted.

λ=1.5μ

The impact of the gaps between the facets on the position measurements will increase for more compact PSFs, i.e. for shorter wavelength. In Figure 3, the errors in the measurement of the PSF position for λ=1.5μ and λ=2.5μ are compared. The black lines in the upper panel are the same as the black line in the upper panel of figure 2, i.e. they present the λ=2.5μ case. The red lines are the same curves for λ=1.5μ. In the lower panel, these curves have been re-computed using only a 3x3 pixel region around the brightest pixel.

figure 3

Conclusion

The simulations show that the center of gravity of acquisition stars can be estimated from a 3x3 region around the brightest pixel with errors less than ±15mas for very high s/n objects. This is only slightly less accurate than the required accuracy of about 10mas.

Calculations similar to the ones shown here could be used to improve the position estimate of targets by taking into account the mapping of the positions of the gaps between facets onto the detector. Even a moderately successful algorithm which predict the offset between measured and true cog with an accuracy of only 50% would suffice to achieve the required accuracy of 10mas for the position estimates.