Optimal transformations for optical multiplex measurements in the presence of photon noise

Hadamard multiplexing is a measurement strategy that yields best sensitivity improvements over scanning measurements for signal-independent detector noise. The presence of photon noise degrades the performance of Hadamard multiplexing because of the increase of photon noise by the superposition of multiple signals. I derive the reduction of the sensitivity gain of a Hadamard measurement and an upper limit for the gain of any cyclic multiplexing strategy in the presence of photon noise. This upper limit clearly exceeds the reduced Hadamard gain and can be achieved by multiplexing sequences that differ from Hadamard S sequences but also share some similarities with respect to their autocorrelation. Examples of such sequences are given. As the analysis shows, the presence of photon noise limits the gain of multiplexing strategies to a finite value, which depends on the ratio between photon noise and detector noise and cannot be exceeded by increasing the number of multiplexed channels. In addition, only switching multiplex schemes, which superpose either all the light or no light of individual channels, can achieve the upper limit of the gain.