  The availability of high-frequency pulsed emitters in the $2-2.5\,\mu$m
wavelength range paved the way for a wealth of new applications in ultrafast
spectroscopy, free-space and fiber-optical communications, surveillance and
recognition, artificial intelligence, and medical imaging. However, developing
these emerging technologies and their large-scale use depend on the
availability of high-speed, low-noise, and cost-effective photodetectors. With
this perspective, here we demonstrate GeSn photodiodes grown on silicon wafers
featuring a high broadband operation covering the extended-SWIR range with a
peak responsivity of 0.3 A/W at room temperature. These GeSn devices exhibit a
high bandwidth reaching 7.5 GHz at 5 V bias with a 2.6 $\mu$m cutoff
wavelength, and their integration in ultrafast time-resolved spectroscopy
applications is demonstrated. In addition to enabling time-resolved
electro-luminescence at 2.3 $\mu$m, the high-speed operation of GeSn detectors
was also exploited in the diagnostics of ultra-short pulses of a supercontinuum
laser with a temporal resolution in the picosecond range at 2.5 $\mu$m.
Establishing these capabilities highlights the potential of manufacturable GeSn
photodiodes for silicon-integrated high-speed extended-SWIR applications.
