Exploiting separation-dependent coherence to boost optical resolution

The problem of resolving pointlike light sources not only serves as a benchmark for optical resolution but also holds various practical applications ranging from microscopy to astronomy. In this paper, we aim to resolve two thermal sources sharing arbitrary mutual coherence using the spatial mode demultiplexing technique. Our analytical study includes scenarios where the coherence and the emission rate depend on the separation between the sources, and is not limited to the faint sources limit. We consider the fluorescence of two interacting dipoles to demonstrate that the dependence of emission characteristics on the parameter of interest can boost the sensitivity of the estimation and noticeably prolong the duration of information decay.

Figure 1

The scheme of the two point sources resolving. The sources emit light into orthogonal modes ${\widehat{s}}_{1,2}$ with a mutual coherence $\gamma$. The light propagates through the lossy imaging system with transmissivity $\kappa$ and PSF width ${\sigma }_{\mathrm{PSF}}$ before being decomposed into spatial modes $f_m\left(x\right)$ with corresponding field operators ${\widehat{a}}_m$ and subsequently detected.

Figure 2

The normalized sensitivity ${\mathcal{M}}_{Re}=M_{Re}4{\sigma }_{\mathrm{PSF}}^2/2\kappa N$ for different real values of the mutual coherence $\gamma$ and numbers of emitted photons $N$ multiplied by the transmissivity $\kappa$ vs separation $d$.

Figure 3

The normalized sensitivity ${\mathcal{M}}_{Im}=M_{Im}4{\sigma }_{\mathrm{PSF}}^2/2\kappa N$ for different imaginary values of the mutual coherence $\gamma$ and numbers of emitted photons $N$ multiplied by the transmissivity $\kappa$ vs separation $d$.

Figure 4

Objects reflect the light from the common thermal illumination source with finite coherence width ${\omega }_c$.

Figure 5

Normalized sensitivity ${\mathcal{M}}_{{\gamma }_0\left(d\right)}=M_{{\gamma }_0\left(d\right)}4{\sigma }_{\mathrm{PSF}}^2/2\kappa N$ in case of the separation-dependent mutual coherence with Gaussian profile (44).

Figure 6

Dipoles emit partially coherent light due to the interaction.

Figure 7

Properties of the dipoles radiation: the normalized emission rate $\dot{N}$ and the degree of mutual coherence $\gamma$ vs separation $d$. Dashed line, time $t=0.02\tau$; solid line, $t=5\tau$. No dephasing ($\eta =0$).

Figure 8

Normalized sensitivity rate $\dot{\mathcal{M}}$ (52) of two dipoles' separation estimation vs time $t$. Light blue line, ${\sigma }_{\mathrm{PSF}}=5\lambda$; red line, ${\sigma }_{\mathrm{PSF}}=\lambda$; dashed line, noninteracting dipoles.

Figure 9

Normalized sensitivity rate $\dot{\mathcal{M}}$ (52) of two dipoles' separation estimation vs separation $d$. Light blue line, ${\sigma }_{\mathrm{PSF}}=5\lambda$; red line, ${\sigma }_{\mathrm{PSF}}=\lambda$; dashed line, noninteracting dipoles. No dephasing ($\eta =0$).

Figure 10

Normalized total sensitivity ${\mathcal{M}}_{\mathrm{tot}}$ (53) of two dipoles' separation estimation vs separation $d$. Light blue line, ${\sigma }_{\mathrm{PSF}}=5\lambda$; red line, ${\sigma }_{\mathrm{PSF}}=\lambda$; dashed line, noninteracting dipoles. No dephasing ($\eta =0$).

Figure 11

Emission characteristics: normalized emission rate $\dot{N}$ and degree of mutual coherence $\gamma$ at the time moment $t=5\tau$. Red line, dephasing-free model $\eta =0$; light blue line, weak dephasing $\eta =0.2/\tau$; gray line, strong dephasing $\eta =2.0/\tau$.

Figure 12

Normalized total sensitivity ${\mathcal{M}}_{\mathrm{tot}}$ (53) for the model with dephasing. Top inset: Weak dephasing $\eta =0.2/\tau$. Bottom inset: Strong dephasing $\eta =2.0/\tau$. Light blue line, ${\sigma }_{\mathrm{PSF}}=5\lambda$; red line, ${\sigma }_{\mathrm{PSF}}=\lambda$; dashed line, noninteracting dipoles.

Figure 13

Normalized sensitivity rate $\dot{\mathcal{M}}$ of two dipoles' separation estimation vs time $t$ in case of noisy detection ${\dot{N}}_{\text{DC}}=0.5{\dot{N}}_0$.

Figure 14

Normalized total sensitivity ${\mathcal{M}}_{\mathrm{tot}}$ vs separation $d$ for the model with detection noise. Top inset: Low rate of the dark counts. Bottom inset: Intense noise.