As technological advancements have enabled higher efficiency compression ignition (CI) engines (diesel engines) by operating at higher pressures and temperatures, the mechanisms of fuel-air mixing are postulated to change. Thus, there is a greater need for experimental data to develop and verify new phenomena. However, the fast time scales and harsh environments associated with CI engines present many experimental difficulties and requires high quality non-intrusive diagnostics to obtain useful data. The diagnostics itself must be validated prior to its implementation in an engine environment. The purpose of this research is to advance rainbow schlieren deflectometry (RSD) to study turbulent fuel sprays including the effects of supercritical mixing in CI engine environments. This objective is met by developing and applying quantitative RSD, for the first time, to three separate aspects of fuel sprays: 1) turbulent mixing, 2) phase boundaries, and 3) non-ideal gas mixing. First, quantitative RSD is demonstrated for turbulent mixing in a canonical helium jet and validated with Rayleigh scattering data in the literature. Second, RSD is applied to high pressure multi-phase fuel sprays, and new methodologies are developed to distinguish the liquid region from the vapor region and to quantify the in-between region on a probabilistic basis. The results of this study demonstrate that RSD can be used as a single diagnostic to measure both liquid and vapor boundaries. Third, the optical-to-thermodynamic relations associated with refractometry are investigated under non-ideal gas mixing conditions to develop a generalized relationship valid for both ideal or non-ideal gas mixing. The last part of this research develops the theoretical tools to understand and experimentally realize supercritical mixing at diesel conditions. Guided by the theoretical analysis, RSD is applied to compare supercritical versus transcritical mixing at CI conditions using a constant pressure test rig. It is concluded that fully supercritical mixing offers several benefits in diesel engines including faster mixing times (50% higher velocities) and the absence of droplets which cause high levels of particulate emissions. However, achieving fully supercritical mixing is difficult in practice because of the requirements of fuels with low critical temperature and/or substantial fuel preheating.