Advances in fabrication of high-finesse optical resonators hold promise for the development of miniaturized, ultra-sensitive, wide-band optical sensors, based on resonance-shift detection. Many potential applications are foreseen for such sensors, among them highly sensitive detection in ultrasound and optoacoustic imaging. Traditionally, sensor interrogation is performed by tuning a narrow linewidth laser to the resonance wavelength. Despite the ubiquity of this method, its use has been mostly limited to lab conditions due to its vulnerability to environmental factors and the difficulty of multiplexing - a key factor in imaging applications. In this paper, we develop a new optical-resonator interrogation scheme based on wideband pulse interferometry, potentially capable of achieving high stability against environmental conditions without compromising sensitivity. Additionally, the method can enable multiplexing several sensors. The unique properties of the pulse-interferometry interrogation approach are studied theoretically and experimentally. Methods for noise reduction in the proposed scheme are presented and experimentally demonstrated, while the overall performance is validated for broadband optical detection of ultrasonic fields. The achieved sensitivity is equivalent to the theoretical limit of a 6 MHz narrow-line width laser, which is 40 times higher than what can be usually achieved by incoherent interferometry for the same optical resonator.