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Impact of optical aberrations on axial position determination by photometry

Nature Methods volume 15, pages 989–990 (2018)Cite this article

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To the Editor — Recently, Franke et al.1 introduced a way to estimate the axial position of single molecules (temporal radial-aperture-based intensity estimation (TRABI)). To this end, they compared the detected photon count from a TRABI estimation to the estimated count from Gaussian point-spread function (PSF) fitting to the data. Empirically they found that the photometric ratio is ~0.7–0.8 at points close to focus and decreases as the distance from the focal plane increases. Here we explain this reported but unexplained discrepancy and, furthermore, show that the photometric ratio as an indicator for axial position is susceptible even to typical optical aberrations.

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 from 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 root-mean-square (r.m.s.; see Supplementary Fig. 5 for experimentally retrieved aberration coefficients). Simulations using the fitted residual aberrations resulted in photometric ratios that agreed well with experimental values. We found a photometric ratio of 85%, in contrast to the values of ~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 matched those obtained from simulations with added aberrations (Fig. 1c). The maximum value of the photometric ratio in focus, overall shape and values strongly depended on the aberrations, which resulted in curves that were broadened, flattened or made asymmetrical. The amounts of added aberrations used here still represent a lens that is referred to as diffraction limited (the Maréchal diffraction limit is 72 mλ), indicating that these aberration levels and combinations thereof are seen in typical setups. We estimated the effects of these small aberrations on the expected axial-position error by carrying out a comparison to an aberration-corrected calibration and found errors between ±100 and ±200 nm over an 800-nm dynamic range (Supplementary Fig. 6). Sample-induced refractive index mismatch, caused by, for example, the use of an oil-immersion objective in a watery enviroment, leads to spherical aberration but also nonspherical components4 on the same order as what we simulated here. In Fig. 1d we show the axial estimation error for seven noncorrected systems from different vendors and labs (details in the Supplementary Methods), which we found to be on the order of ±50 to ±100 nm. We measured the aberrations in these systems via through-focus bead acquisition (see Supplementary Fig. 7 for details on the individual wavefront errors). We then calculated the axial estimation error as the difference between the calibrated aberration-free photometric-ratio-based position estimate and the simulated estimate with aberrations equaling the experimentally determined microscope aberrations. We conclude that in order for the photometric ratio to be converted to a viable, accurate depth map, the optical aberrations must be known to a very high degree (wavefront uncertainty < 10 mλ results in axial uncertainty < 20 nm).

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Fig. 1: Photon-count estimation and the effects of aberrations on the photometric ratio.

Data availability

The data are available for download at https://doi.org/10.4121/uuid:557b6445-5d40-402a-b214-93d7c6415195. The software is available as open-source Matlab scripts from ftp://qiftp.tudelft.nl/rieger/outgoing/Rasmus_photoncount.zip.

References

  1. Franke, C., Sauer, M. & van de Linde, S. Nat. Methods 14, 41–44 (2017).

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  4. Burke, D., Patton, B., Huang, F., Bewersdorf, J. & Booth, M. J. Optica 2, 177–185 (2015).

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Acknowledgements

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

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Author notes

  1. These authors contributed equally: Sjoerd Stallinga, Bernd Rieger.

Authors and Affiliations

  1. Department of Imaging Physics, Delft University of Technology, Delft, The Netherlands

    Rasmus Ø. Thorsen, Christiaan N. Hulleman, Sjoerd Stallinga & Bernd Rieger

  2. RNA Therapeutics Institute, University of Massachusetts Medical School, Worcester, MA, USA

    Mathias Hammer & David Grünwald

Authors
  1. Rasmus Ø. Thorsen
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  2. Christiaan N. Hulleman
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  3. Mathias Hammer
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Contributions

R.Ø.T. performed simulations and analyzed data. C.N.H. performed experiments. M.H. and D.G. provided 3D PSF data from several microscopes. S.S. and B.R. designed and coordinated the research. B.R., S.S. and R.Ø.T. wrote the manuscript, and all authors commented on it.

Corresponding authors

Correspondence to Sjoerd Stallinga or Bernd Rieger.

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The authors declare no competing interests.

Integrated supplementary information

Supplementary Figure 1 Reproducibility of photon count estimation.

(A-C) Estimated photon counts and background photons per pixel for three measurements of three separated 45-nm-diameter beads imaged with an aberration-corrected microscope as a function of analysis area on the camera; pixel size, 80 nm. The three lines show the count for fitting with a fully fledged vectorial, TRABI or a simple Gaussian PSF.

Supplementary Figure 2 Comparison of photon count estimations in simulation and experiment.

(A) Photon count estimation for a single simulated 45-nm bead PSF as a function of PSF image area on the camera for three different estimation algorithms: vectorial fit, TRABI, and Gaussian fit (8,750 signal photons, 15 background photons, same pixel size, NA and refractive index values as experiment). (B) The photometric ratio between Gaussian fit and TRABI photon count estimation as a function of PSF fit size. The dashed line indicates when the circular aperture size fits precisely within the square region of interest. (C-D) Same as panels (A,B) but for an experimentally recorded aberration-corrected 45-nm bead, indicating good quantitative agreement between experiment and simulation.

Supplementary Figure 3 TRABI photon count estimate as a function of aperture size and PSF model.

(A) Ratio of the estimated photon count by TRABI and simulated number of signal photons captured at the camera as a function of aperture diameter for two PSF models. TRABI aperture diameter (d) of 2 × 1.86 × FWHM (FWHM = 214.5 nm) as suggested in Franke et al.1 is indicated. For this aperture size, we find TRABI to estimate 92% and 85% of the total number of photons according to the low-NA scalar Airy PSF model and the high-NA vectorial PSF model, respectively. The fraction of the total energy contained within an aperture of the prescribed radii in the Airy PSF model (PSFAiry(ρ) = [2J1(D)/(D)]2 where D = 2πρ(NA/λ); FWHM = 0.514(λ/NA)) follows the analytically derived formula by Born & Wolf (ref. 4 in the Supplementary Methods reference list; Chapter 8; Fig. 8.13), in contrast to the 100% reported by Franke et al.1 in their supplement. The PSF simulations were performed with 2,500 signal photons and no background photons and a pixel size of 1 nm (to exclude quantization effects). Otherwise, the simulation parameters were the same as the experimental parameters. (B-C) Characterization of the Airy and vectorial PSFs as a function of the radial position in a linear and log scale, respectively, showing the long tail of the PSF.

Supplementary Figure 4 The effect of bead size on PSF shape.

(A) The lateral and axial FWHM as a function of bead size. (B-C) Lateral average PSF in linear and log scale, respectively, as a function of the radial positon. The PSFs are displayed for different bead sizes. (D-E) On-axis PSF in linear and log scale, respectively, as a function of the axial position. The 45-nm bead captures the full tail behavior in both the lateral and axial directions of an actual single-molecule emitter, and has an FWHM within a few nanometers of a single-molecule emitter, whereas PSF details clearly get lost with larger (>90 nm) beads.

Supplementary Figure 5 Quantification of aberration retrieval and correction.

(A) Through-focus PSF image stacks of experimental and fitted PSFs after aberration correction and, subsequently, aberrated with a single primary Zernike mode: astigmatism (\({\mathrm{Z}}_2^2\)), coma (\({\mathrm{Z}}_3^1\)), and spherical aberration (\({\mathrm{Z}}_4^0\)), with aberration coefficients 36 mλ and 72 mλ (root-mean-square values). The region of interest for each PSF image is 31 × 31 pixels with a pixel size of 80 nm. All 4 × 4 sub-image pairs are contrast-stretched with the same factor for better visibility of spot shape. The estimated photon counts were within 9,500–21,000 signal photons and 15–18 background photons per pixel. Reproducibility as shown in (B). (B) Fitted Zernike modes and retrieved aberration coefficients. The coefficients are averaged over six measurements with error bars indicating one s.d. The aberration fit routine includes all tertiary Zernike modes (all \({\mathrm{Z}}_{\mathrm{n}}^{\mathrm{m}}\) with 2<n+|m|≤8) and assumes optical parameters as described in the Supplementary Methods. (Horizontal dashed lines are used to guide the eye.)

Supplementary Figure 6 Axial calibration of the photometric ratio and axial estimation error caused by small single-mode aberrations.

(A) The photometric ratio over six 45-nm-bead measurements after aberration correction (Exp.) and simulated vectorial PSFs (Sim.) as a function of axial position (see Supplementary Methods). (B) The estimated axial position of aberration-corrected beads as a function of the calibrated axial position shown in (A). Around focus the error is a bit larger as expected owing to the near-parabolic shape of the photometric ratio curve (see Supplementary Methods for a description of the error analysis). (C) Estimated axial position for single-mode aberrations on 45-nm beads using the calibrated axial position from the aberration-corrected data in (A). The gray dashed line is used to guide the eye for the correct estimation. Experimental data (Exp.) in A,B,C are shown as mean ± s.d. over six bead measurements. (D) Error of the estimated axial position as the distance between the mean estimated position and the true (calibrated) position.

Supplementary Figure 7 Typical microscope aberrations and their influence on the estimated axial position.

(A) Fitted Zernike modes and retrieved aberration coefficients for several microscopes, Data 1–7 (specifications are given in the Supplementary Methods). The coefficients are averaged over six and one bead measurement(s) for datasets 1–6 and 7, respectively, with error bars indicating one s.d., and WRMS is the mean of the wavefront error. The aberration fit routine includes all tertiary Zernike modes and assumes optical parameters for each microscope as described in the Supplementary Methods. (B) Estimated axial position for simulated single-molecule PSFs with aberrations equaling the mean experimentally found microscope aberrations in (A) compared to the calibrated axial position using the aberration-corrected photometric ratio (see Supplementary Fig. 6a). Area of fit 7 × 7 pixels and aperture radius 1.86 × FWHM.

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Thorsen, R.Ø., Hulleman, C.N., Hammer, M. et al. Impact of optical aberrations on axial position determination by photometry. Nat Methods 15, 989–990 (2018). https://doi.org/10.1038/s41592-018-0227-4

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