Intensity-Based Axial Localization at the Quantum Limit

We derive fundamental precision bounds for single-point axial localization. For Gaussian beams, this ultimate limit can be achieved with a single intensity scan, provided the camera is placed at one of two optimal transverse detection planes. Hence, for axial localization there is no need of more complicated detection schemes. The theory is verified with an experimental demonstration of axial resolution 3 orders of magnitude below the classical depth of focus.

Figure 1

Scheme of the axial localization experiment with a relay optical system.

Figure 2

Fisher information at the image space about axial (blue solid line) and transversal (red broken line) positions of the object waist in units of their respective QFI for different placements of the detector. We use $z=5$, $f=1$, $z_R=1$, $w_0=1$ and the detector positions are relative to the beam waist in units of $z_R^{\prime }$.

Figure 3

Experimental setup used to measure the axial displacement $z$. See text for a detailed description.

Figure 4

Normalized radial density of Fisher information (solid blue line) and normalized beam intensity profile (dashed red line) in the optimal detection plane, where the beam has a width $w_{\mathrm{opt}}^{\prime }$. The system parameters are the same as in Fig. 2.

Figure 5

Experimental estimation of axial displacements ${\delta }_z$ from the nominal object plane with respect to which the camera is optimally placed (9). The inset shows the statistics of the estimator ${\widehat{\delta }}_z$ as defined in Eq. (10). In the main plot the true distance was subtracted from the estimates to get a more convenient scale on the vertical axis. The blue strips depict the quantum bound for the $2\times {10}^6$ detections and $z_R=18.9\mu \mathrm{m}$.