In theory, an optical system with a finite aperture can be coated to produce arbitrarily perfect imagery over a limited field. When the object is of limited extent, this field can be made the optical conjugate to the object, so that the whole object is imaged with arbitrary precision. The required pupil coating approximates low-contrast cosine fringes over its central region, with a frequency and amplitude that rapidly accelerate as the aperture edge is approached. Here the maximum occurs as a narrow spike. The frequency near the central region varies directly with the total extent of the conjugate field, and inversely with the required central core width Δ in the point amplitude response. As Δ is made arbitrarily narrow, the point amplitude response approaches the form of a sinc function over the field of view. This function is precisely the point amplitude for a diffraction-limited pupil with a magnified aperture of 1/Δ times the given pupil aperture ! The only image property that is not in compliance with this effective aperture magnification is that of total illumination. This is severely reduced from that of the original, uncoated aperture, and is the major restriction on practical use of the derived pupil. Applications to microscopy and telescopy are discussed.