HIRES, a high resolution spectrometer, is one of the first five instruments foreseen in the ESO roadmap for the E-ELT. This spectrograph should ideally provide full spectral coverage from the UV limit to 2.5 microns, with a resolving power from R$sim$10,000 to R$sim$100,000. At visual/blue wavelengths, where the adaptive optics (AO) cannot provide an efficient light-concentration, HIRES will necessarily be a bulky, seeing-limited instrument. The fundamental question, which we address in this paper, is whether the same approach should be adopted in the near-infrared range, or HIRES should only be equipped with compact infrared module(s) with a much smaller aperture, taking advantage of an AO-correction. The main drawbacks of a seeing-limited instrument at all wavelengths are: textit{i)} Lower sensitivities at wavelengths dominated by thermal background (red part of the K-band). textit{ii)} Much higher volumes and costs for the IR spectrograph module(s). The main drawbacks of using smaller, AO-fed IR module(s) are: textit{i)} Performances rapidly degrading towards shorter wavelengths (especially J e Y bands). textit{ii)} Different spatial sampling of extended objects (the optical module see a much larger area on the sky). In this paper we perform a trade-off analysis and quantify the various effects that contribute to improve or deteriorate the signal to noise ratio. In particular, we evaluate the position of the cross-over wavelength at which AO-fed instruments starts to outperform seeing-limited instruments. This parameter is of paramount importance for the design of the part of HIRES covering the K-band.
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