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We study the dynamic structure factor of a one-dimensional Bose gas confinedin an optical lattice and modeled by the Bose-Hubbard Hamiltonian, using avariety of numerical and analytical approaches. The dynamic structure factor,experimentally measurable by Bragg spectroscopy, is studied in three relevantcases: in the clean regime, featuring either a superfluid or a Mott phase; andin the presence of two types of (quasi-)disordered external potentials: aquasi-periodic potential obtained from a bichromatic superlattice and arandom-box disorder - both featuring a Bose glass phase. In the clean case, weshow the emergence of a gapped doublon mode (corresponding to a repulsivelybound state) for incommensurate filling, well separated from the low-energyacoustic mode. In the disordered case, we show that the dynamic structurefactor provides a direct insight into the spatial structure of the excitations,unveiling their localized nature, which represents a fundamental signature ofthe Bose glass phase. Furthermore, it provides a clear fingerprint of the verynature of the localization mechanism which differs for the two kinds ofdisorder potentials we consider. In special cases, the dynamic structure factormay provide an estimate of the position of the localization transition fromsuperfluid to Bose glass, in a complementary manner to the information deducedfrom the momentum distribution.

Bragg spectroscopy of clean and disordered lattice bosons in one dimension: a spectral fingerprint of the Bose glass