A prototypical experiment in cavity quantum electrodynamics involves controlling the light-matter interaction by tuning the frequency of a cavity mode in- and out-of resonance with the frequency of a quantum emitter,1-3 while the field amplitude is generally unaltered. The opposite situation, where one perturbs the spatial pattern of a cavity mode without changing its frequency, has been considered only recently in a few works.4, 5 Changing the amplitude of the field at the emitter's position has important applications, at it allows a real-time control of the light-matter coupling rate, and therefore a direct control of processes such as spontaneous emission and Rabi oscillations. In view of this large potential, in this paper we discuss general design principles that allow obtaining large variations of the electromagnetic field, without change of the frequency, upon an external perturbation of the cavity. We showcase the application of these rules to two photonic structures, a single Fabry-Perot cavity and a coupled three-cavity system. As showed by our analysis and by the examples provided, a small frequency spacing between the modes of the unperturbed cavity is an important requirement to obtain large field variations upon small perturbations. In this regard, a coupled-cavity system, where the frequency spacing is controlled by the interaction rates between the single cavities, constitutes the most promising system to achieve large modulations of the field amplitude.