Scaling with telescope size

The $d\lambda \propto D_{tel}/D_{pupil}$ scaling means that large telescopes require instruments with large pupils - and that means bigger collimators, cameras, gratings, and detectors. Note that this is true even for a single-object spectrograph where we don't need to image a large field of view, but want a reasonably high resolution.

Constructing large collimators and cameras is a challenge and constructing very large diffraction gratings is extremely difficult. The only way to get around needing a large pupil is to have a narrower slit. That means either suffering slit losses when the slit gets smaller than the seeing, or improving the image resolution delivered to the telescope focal plane, such as with adaptive optics. This is one reason that extremely large telescopes will need and use adaptive optics - to keep some of the instruments to a buildable size.

Taking an instrument from a small telescope and putting it on a big telescope can be done (if the telescopes have the same f-number - if the f-number is different, it will either underfill or overfill the pupil). However, it is not ideal. The larger telescope has a larger scale at its focal plane, so for e.g. a 1.0'' slit we need a physically wider slit. With the wider slit, the resolution is worse because we are allowing a larger range of angles onto the grating, just as it would be if we sat at the small telescope and opened the slit from 1'' to 2''.

If we just want to do imaging, we can move a reimaging camera from a small to large telescope, but of course its field of view will be proportionately smaller. The product of $D_{tel}^2 * (field diameter)^2$ stays constant, so if we need to map an area larger than the field of view, the large telescope won't be faster - unless it has better image quality.

A number of instruments have used a multi-barrel reimager strategy to cover larger areas. The idea is that for a 2x larger diameter telescope, instead of scaling up a instrument design by 2x diameter (8x the volume of the small instrument, 4x the number of detector pixels), one builds 4 of the smaller spectrographs and tiles the focal plane with them. This covers the same field as the one big spectrograph, with the same number of pixels, and theoretically requires only 4x the volume of the small instrument. Although one has to make 4 of each optical element, the elements are all smaller, so it should be cheaper and less challenging to fabricate. But a drawback is that they are each still using a small pupil, so the resolution of each spectrograph will be limited: for a given slit and grating, the resolution will be 2x worse than it was on the small telescope.

Acknowledgments

My understanding of astronomical instruments and their design considerations has benefitted greatly from conversations with Ted Williams, Steve Shectman, and Rebecca Bernstein. Sylvain Veilleux and Alan Dressler gave me the opportunity to work on an instrument for a large telescope. During the writing of this document, I have been supported by NASA/Spitzer contract 1255094 issued by JPL/Caltech.