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# Impact of optical aberrations on axial position determination by photometry

Nature Methodsvolume 15pages989990 (2018) | Download Citation

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## 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. 1.

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

2. 2.

Siemons, M., Hulleman, C. N., Thorsen, R. Ø., Smith, C. S. & Stallinga, S. Opt. Express 26, 8397–8416 (2018).

3. 3.

Stallinga, S. & Rieger, B. Opt. Express 18, 24461–24476 (2010).

4. 4.

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

## 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.

## Author information

### Author notes

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

### 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

### 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.

### Competing interests

The authors declare no competing interests.

### Corresponding authors

Correspondence to Sjoerd Stallinga or Bernd Rieger.

## Integrated supplementary information

1. ### 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.

2. ### 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.

3. ### 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.

4. ### 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.

5. ### 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.)

6. ### 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.

7. ### 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.

## Supplementary Information

1. ### Supplementary Text and Figures

Supplementary Figures 1–7 and Supplementary Methods

### DOI

https://doi.org/10.1038/s41592-018-0227-4