## Abstract

Recently, Franke, Sauer and van de Linde^{1} introduced a way to estimate the axial position of single-molecules (TRABI). To this end, they compared the detected photon count from a temporal radial-aperture-based intensity estimation to the estimated count from Gaussian point-spread function (PSF) fitting to the data. Empirically they found this photometric ratio to be around 0.7-0.8 close to focus and decreasing away from it. Here, we explain this reported but unexplained discrepancy and furthermore show that the photometric ratio as indicator for axial position is susceptible even to typical optical aberrations.

In Fig. 1A we show the photon count from a 45 nm bead imaged with an aberration-corrected microscope^{2} (see **Supplementary Methods** for details) estimated by three different methods (Gaussian PSF fit, TRABI, Vectorial PSF fit^{3}) as a function of aperture radius or fit box size, respectively (for reproducibility see Supplementary Fig. 1). It is evident that the estimated count increases with increasing area for all three methods, i.e. no method finds the true count for a realistic area as the true microscope PSF has a very long tail.Simulations of full-vectorial PSFs support this conclusion (Supplementary Fig. 2), showing that the tail deviates substantially from the Airy PSF model^{3}. It is also evident that with any aperture based method the true count can only be approximated up to 90% with aperture radii less than one micron (Supplementary Fig. 3) and that Gaussian PSF fitting performs worse as a Gaussian cannot fit the long tail at all. This, however, does not bias the localization estimate of Gaussian fitting for round spots^{3}. The suitability of sub-diffraction sized beads for these experiments was investigated in simulation and found to increase the FWHM by only a few nanometers compared to the single-molecule PSF (Supplementary Fig. 4) while giving access to more light over a longer period during the experiment.

Next, we varied the axial position of the sample while imaging aberration corrected beads and evaluated the photometric ratio between photon count estimates from Gaussian fitting and TRABI as a function of defocus, as shown in Fig. 1B (see Supplementary Fig. 2 for sensitivity to fit area). The residual wavefront aberration was 24 mλ RMS (see Supplementary Fig. 5 for experimentally retrieved aberration coefficients). Simulations using the fitted residual aberrations result in photometric ratios that agree well with experiment. We find a photometric ratio of 85% in contrast to the values around 75% in focus reported by Franke et al.^{1}, which we attribute to aberrations present in their experiment. To assess the influence of aberrations, we experimentally engineered PSFs with small amounts of astigmatism, coma or spherical aberration. Photometric ratios obtained from these experiments match those obtained from simulations with added aberrations (see Fig. 1C). The maximum value of the photometric ratio in focus, overall shape and values strongly depend on the aberrations, resulting in curves that are either broadened, flattened or made asymmetrical. The amounts of added aberrations still represent a lens that sells as diffraction limited (Maréchal diffraction limit is at 72 mλ), indicating that these aberration levels and combinations thereof are seen in typical setups. We estimated the impact of these small aberrations on the expected axial position error by comparing against an aberration corrected calibration and find errors between ±100 to ±200 nm over 800 nm dynamic range (see Supplementary Fig. 6). We inspected seven different setups for aberrations and found that typical non-corrected systems can have axial errors on the order of ±50 to ±100 nm (see Supplementary Fig. 7). Sample induced refractive index mismatch, e.g. by using oil immersion into a watery enviroment, leads to spherical aberration but also non-spherical components^{4} on the same order as we simulated here. We conclude that in order to convert the photometric ratio to a viable, accurate depth map the optical aberrations must be known to a very high degree (wave front uncertainty < 10 mλ results in axial uncertainty < 20 nm).

## Data availability

The data is available for download at https://data.4tu.nl/download/uuid:ea2ea179-26f4-4e1a-90e1-b2759b553ce8/. The software is available as Matlab scripts in open-source from ftp://qiftp.tudelft.nl/rieger/outgoing/Rasmus_photoncount.zip

## Author contribution

RØT performed simulations and analyzed data, CNH performed experiments, MH and DG provided 3D PSF data from several microscopes, SS and BR designed and coordinated the research. BR, SS, RØT wrote the manuscript, and all authors commented on it.

## Competing interests

The authors declare no competing financial interests.

## Acknowledgments

B.R., C.N.H. acknowledge European Research Council grant no. 648580 and B.R., R.Ø.T., D.G. acknowledge National Institute of Health grant no. U01EB021238. We thank Job Dekker for providing access to serval microscopes and Keith Lidke for providing PSF data.