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Diffraction-limited spectroscopy with adaptive optics (AO) has several advantages over traditional seeing-limited spectroscopy. First, high resolution can be achieved without a large loss of light at the entrance slit of the spectrograph. Second, the small AO image width allows the cross-dispersed orders to be spaced closer together on the detector, allowing a large wavelength coverage. Third, AO spectrograph optics are slow and small, costing much less than for a traditional spectrograph. Fourth, small AO images provide high spatial resolution. Fifth, scattered light is less problematic. And last, the small entrance slit of the spectrograph can get rid of much of the sky background to obtain spectra of faint objects. We have done theoretical calculations and simulations for infrared spectroscopy at the MMT 6.5 m with laser guide star AO, which provides almost full sky coverage. The results show we can expect 40-60% of the photons from a unresolved source within 0.2 arcsec diameter circle for J, H, K, L and M bands under typical atmospheric seeing condition at 2.2 micron (ro = 1.0 m, to = 21 ms, ?o = 15 arcsec and d0 = 25 m). Therefore, the spectrograph entrance slit size should match the 0.2 arcsec image to obtain high throughput. Higher resolution can be achieved by narrowing down the slit size to match the diffraction-limited image core size of about 0. 1 arcsec in the infrared. However, the throughput will be correspondingly reduced by a factor of two. Due to the limited atmospheric isoplanatic angle in the J, H and K bands, the encircled photon percentage within 0.2 arcsec diameter drops from 40-60% when the object is at the laser pointing direction to 20-40% when the object is about 30 arcsec away from the laser direction. Therefore, the useful field of view for AO multiple object spectroscopy is about 60 arcsec. Further studies of JR background (sky and thermal) and JR detector performance show that spectral resolution of R = 2,000 can take full advantage of AO images without much penalty due to the dark current of the JR detector and JR OH sky emission lines. We have also studied natural guide star AO spectroscopy. Though sky coverage for this kind of spectroscopy at the MMT 6.5 m is very limited, a bright star provides much better performance than the laser guide star AO spectroscopy. About 40-70% photons are concentrated within 0.1 arcsec diameter for guide stars brighter than 13 magnitude. Therefore, higher resolution and high throughput can be obtained simultaneously, given a bright enough natural guide object. The field-of-view for multiobject spectroscopy using a natural guide star is similar to that for laser guiding.
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