The nature of phase and amplitude modulation in heterodyne background-oriented schlieren systems.

Heterodyne background-oriented schlieren is examined here as a diagnostic that separates phase and amplitude effects in reactive flows, and it is applied to a lean premixed swirl-stabilised methane flame. In this approach, a periodic fringe background and Fourier demodulation yield line-of-sight displacement fields associated with refractive-index gradients, while the fringe amplitude modulation isolates intensity redistribution in a manner that is mathematically decoupled from phase. Building on this separation, ray-tracing analysis clarifies how the optical system and the refractive-index field jointly shape the recorded image, thereby revealing the physical origin of the fringe amplitude modulation signal. In particular, the fringe amplitude modulation field is shown to behave similarly to a focussed shadowgraph that preferentially amplifies high-gradient, small-scale features linked to combustion and suppresses low-curvature convection features. This selectivity is governed by the acquisition aperture through depth-of-focus and aperture-related ray blockage. Finally, controlled experiments with extensive image acquisition corroborate these interpretations and provide practical guidance on carrier-filtering and acquisition settings to avoid spectral overlap and maintain accuracy. Graphical abstract: Supplementary Information: The online version contains supplementary material available at 10.1007/s00348-026-04289-w.