  Recently an optical amplification process called the plasmon injection scheme
was introduced as an effective solution to overcoming losses in metamaterials.
Implementations with near-field imaging applications have indicated substantial
performance enhancements even in the presence of noise. This powerful and
versatile compensation technique, which has since been renamed to a more
generalized active convolved illumination, offers new possibilities of
improving the performance of many previously conceived metamaterial-based
devices and conventional imaging systems. In this work, we present the first
comprehensive mathematical breakdown of active convolved illumination for
coherent imaging. Our analysis highlights the distinctive features of active
convolved illumination, such as selective spectral amplification and
correlations, and provides a rigorous understanding of the loss compensation
process. These features are achieved by an auxiliary source coherently
superimposed with the object field. The auxiliary source is designed to have
three important properties. First, it is correlated with the object field.
Second, it is defined over a finite spectral bandwidth. Third, it is amplified
over that selected bandwidth. We derive the variance for the image spectrum and
show that utilizing the auxiliary source with the above properties can
significantly improve the spectral signal-to-noise ratio and resolution limit.
Besides enhanced superresolution imaging, the theory can be potentially
generalized to the compensation of information or photon loss in a wide variety
of coherent and incoherent linear systems including those, for example, in
atmospheric imaging, time-domain spectroscopy, ${\cal PT}$ symmetric
non-Hermitian photonics, and even quantum computing.
