  High-bandwidth, fiber-based optical cavities are a promising building block
for future quantum networks. They are used to resonantly couple stationary
qubits such as single or multiple atoms with photons routing quantum
information into a fiber network at high rates. In high-bandwidth cavities,
standard fluorescence imaging on the atom-cavity resonance line for controlling
atom positions is impaired since the Purcell effect strongly suppresses
all-directional fluorescence. Here, we restore imaging of $^{87}$Rb atoms
strongly coupled to such a fiber Fabry-P\'erot cavity by detecting the repumper
fluorescence which is generated by continuous and three-dimensional Raman
sideband cooling. We have carried out a detailed spectroscopic investigation of
the repumper-induced differential light shifts affecting the Raman resonance,
dependent on intensity and detuning. Our analysis identifies a compromise
regime between imaging signal-to-noise ratio and survival rate, where physical
insight into the role of dipole-force fluctuations in the heating dynamics of
trapped atoms is gained.
