Quantum key distribution is rising as an important cryptographic primitive for protecting the communication infrastructure in the digital era. However, its implementation security is often weakened by components whose behavior deviates from what is expected. Here we analyze the response of a self-differencing avalanche photodiode, a key enabler for high speed quantum key distribution, to intense light shone from a continuous-wave laser. Under incorrect settings, the cancellation entailed by the self-differencing circuitry can make the detector insensitive to single photons. However, we experimentally demonstrate that even in such cases intensity modulation can be used as an effective measure to restore the detector's expected response to the input light.