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

Figure 1

Schematic of PSI. Photons are input into the interferometer. The phase (${\varphi }_{\mathrm{m}}$) and transmittance ($T$) for the sample are measured through the detection of the photons with single-photon detectors (SPD1 and SPD2) at the output. The phase shifter (PS) is used to control the phase of the interferometer. BS: beam splitter. ${\theta }_{\mathrm{off}}$: offset phase of PS. $\theta$: phase shift given by PS.

Figure 2

Measurement uncertainties in the estimation using PSI for $N=100$ input photons. [(a)–(d)] Uncertainties in $T$. Panels (a) and (c) illustrate $\mathrm{\Delta }{T}_{\mathrm{PS}}$ and $\mathrm{\Delta }{T}_{{\mathrm{PS}}^{\prime }}$, respectively. In (b) the orange dashed and red dot-dashed curves indicate $\mathrm{\Delta }{T}_{\mathrm{PS}}$ with $cos\left(4\varphi \right)=1$ and $cos\left(4\varphi \right)=-1$, respectively. In (d) the orange dashed and red dot-dashed curves indicate $\mathrm{\Delta }{T}_{{\mathrm{PS}}^{\prime }}$ with $cos\left(4\varphi \right)=1$ and $cos\left(4\varphi \right)=-1$, respectively. The blue solid curves in (b) and (d) indicate $\mathrm{\Delta }{T}_{\mathrm{S}}$. [(e), (f)] Uncertainties in $\varphi$. Panel (e) illustrates $\mathrm{\Delta }{\varphi }_{\mathrm{PS}}$. In (f) the orange dashed and red dot-dashed curves indicate $\mathrm{\Delta }{\varphi }_{\mathrm{PS}}$ with $cos\left(4\varphi \right)=1$ and $cos\left(4\varphi \right)=-1$, respectively. The blue solid curve indicates $\mathrm{\Delta }{\varphi }_{\mathrm{S}}$.

Figure 3

Measurement uncertainties in the estimation with PSI for $N=100$ input photons which have a Poisson fluctuation. (a) Uncertainties in $T$. The red dashed curve, the purple dotted curve, and the blue solid curve indicate $\mathrm{\Delta }{T}_{\mathrm{PSP}}, \mathrm{\Delta }{T}_{{\mathrm{PSP}}^{\prime }}$, and $\mathrm{\Delta }{T}_{\mathrm{S}}$, respectively. (b) Uncertainties in $\varphi$. The red dashed curve and the blue solid curve indicate $\mathrm{\Delta }{\varphi }_{\mathrm{PS}}$ and $\mathrm{\Delta }{\varphi }_{\mathrm{S}}$, respectively.