Precision limit for simultaneous phase and transmittance estimation with phase-shifting interferometry

The use of phase-shifting interferometry (PSI) in the ultra-low-light region would open way for important applications, such as imaging phototoxic substances and sensing ultrafast phenomena. In region, statistical nature photons can be a dominant source noise compared with other caused by experimental instruments. Here we theoretically derive precision limit determined simultaneously measuring transmittance phase sample PSI. We show that PSI depends on themselves. also trade-off relation between phase. Then compare sequential optimal measurements which are separately measured each corresponding measurement. discuss case where input number fluctuates Poisson distribution fluctuation affects both