The proposed Giant Magellan Telescope (GMT) has a number of features that are well-suited to the task of imaging extrasolar planets in nearby star systems. The principal aid to this task is the large clear aperture segments which are relatively easy to apodize. This paper considers the methods currently envisaged to be of practical use for the task. In addition to star and planet fluxes, exoplanet imaging is dependent on aperture, throughput, bandwidth, beamwidth (FWHM), Strehl ratio (SR), and halo structure. Adaptive optics systems increase the SR, simultaneously dropping the residual scattered halo. This reveals the diffracted halo, which now becomes the limiting factor. Apodization reduces the diffracted halo, but at a cost in terms of throughput and a corresponding increase in photon noise. Since the best known ideal apodizations also have very low throughputs, they are not the best choices for ground-based exoplanet imaging. In addition, the ultra-low diffracted halos from these apodizations provide no benefit below the residual scattered halo, which is not helped by apodization. We consider instead a family of apodizations that have sufficiently dark diffracted halos, while retaining relatively high throughputs. One form of apodization
can be applied to the GMT pupil using replicated apodization of individual segments, providing a low-halo survey mode that is high throughput and matched to the AO system. Since the reduced halo from the apodized segments only allows high-contrast detection to within a few λ/Dsegment of the star, the single segment methods are limited by the segment size. We also consider the potential for apodizing the full aperture for high contrast at a few times λ/Dfull through the use of an applied phase pattern, using either the adaptive secondary or a separate phase mask. We conclude that the phase mask method offers the best advantage for S/N since it does not lose light like the apodization schemes. However, it does have a restricted azimuthal search area, requiring multiple exposures to complete a survey. It appears to be the clearly best method for examining the exoplanet once discovered. It should be possible to apply offsets to the GMT's adaptive secondary to achieve constrasts of 10-5 at 2xFWHM (27 mas at 1.65 μm, 80 mas at 5 μm) and 10-6 at 3xFWHM (42 mas at 1.65 μm, 200 mas at 5 μm).