Single molecule light field microscopy

Ruth R Sims, Sohaib Abdul Rehman, Martin O Lenz, Sarah I Benaissa, Ezra Bruggeman, Adam Clark, Edward W Sanders, Aleks Ponjavic, Leila Muresan, Steven F Lee, et al.

► To cite this version:

Ruth R Sims, Sohaib Abdul Rehman, Martin O Lenz, Sarah I Benaissa, Ezra Bruggeman, et al. Single molecule light field microscopy. Optica, 2020, 7 (9), pp.1065. 10.1364/OPTICA.397172. hal-05451495

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# Single molecule light field microscopy

RUTH R. SIMS,<fnref n="1" />,<fnref n="2" />,† SOHAIB ABDUL REHMAN,<fnref n="1" />,† ID MARTIN O. LENZ,<fnref n="1" /> ID SARAH I. BENAISSA,<fnref n="1" /> EZRA BRUGGEMAN,<fnref n="3" /> ID ADAM CLARK,<fnref n="3" /> EDWARD W. SANDERS,<fnref n="3" /> ID ALEKS PONJAVIC,<fnref n="3" />,<fnref n="4" />,<fnref n="5" /> LEILA MURESAN,<fnref n="1" /> STEVEN F. LEE,<fnref n="3" /> AND KEVIN O'HOLLERAN<fnref n="1" />,\* ID

Received 13 May 2020; revised 12 July 2020; accepted 12 July 2020 (Doc. ID 397172); published 21 August 2020

We introduce single molecule light field microscopy (SMLFM), a new class of three-dimensional (3D) single molecule localization microscopy. By segmenting the back focal plane of a microscope objective with an array of microlenses to generate multiple 2D perspective views, the same single fluorophore can be imaged from different angles. These views, in combination with a bespoke fitting algorithm, enable the 3D positions of single fluorophores to be determined from parallax. SMLFM achieves up to 20 nm localization precision throughout an extended 6 µm depth of field. The capabilities of SMLFM are showcased by imaging membranes of fixed eukaryotic cells and DNA nanostructures below the optical diffraction limit.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

https://doi.org/10.1364/OPTICA.397172

## 1. INTRODUCTION

Single molecule localization microscopy (SMLM) has emerged as one of the most popular approaches to superresolution fluorescence imaging, in part due to the relative simplicity of its experimental implementation [1]. SMLM achieves super-resolution through localization of the emitter position with subdiffraction limited precision in sparsely fluorescing specimens. In three-dimensional (3D) SMLM, axial information is usually obtained at the detriment of both precision and resolution resulting from a reduction in photon throughput (attributable to additional optical elements); intrinsically higher background from out-of-focus emitters; and extended point spread functions (PSFs) resulting in overlapping images of different emitters and ultimately a degradation in precision [2]. We introduce single molecule light field microscopy (SMLFM), a simple and highly efficient 3D superresolution imaging technique that combines the complementary strengths of SMLM and light field detection to achieve superresolution imaging throughout a continuous 3D volume.

Numerous approaches to extend the depth of field (DOF) of SMLM have been developed [3–6]. The most successful examples perform single-shot 3D imaging by modifying the shape of the intensity PSF to encode axial information. Astigmatic or rotating double-helix PSFs have DOFs ranging from 0.5 to 4 µm [7,8]. Self-bending PSFs such as Airy beams have also been used to encode the axial position in SMLM [9]. Recently, asymmetric

Airy beam approaches have improved the technique's axial range to 7 µm [10]. Large axial ranges have been achieved using other wavefront engineering approaches [11], such as Tetrapod (Saddle Point) [12] and secondary astigmatism [13]. However, in these cases, extracting the super-resolved positions of single molecules is generally more challenging than in 2D SMLM because the engineered PSFs (with the exception of the Airy beam) cannot accurately be approximated by 2D Gaussian functions. As a result, computationally expensive, maximum-likelihood approaches, that necessitate phase retrieval [14,15] or spline fitting [16] to generate finely tuned templates that account for system- and sample-induced aberrations are required. Furthermore, the large spatial footprint of these sculpted PSFs decreases the maximum achievable localization density, significantly decreasing throughput. Another approach called multifocal plane microscopy (MPM) images a discrete number of axial planes to different lateral positions on one or more detectors [17–20], and can be combined with the aforementioned PSF engineering approaches. However, since an emitter located on a particular axial plane is in focus in a subset of these images, while contributing background to the others, only a fraction of the total number of detected photons contribute to the precision of each localization.

Light field microscopes based on refractive microlens arrays (MLAs) have PSFs composed of an array of spots, each of which resembles a 2D Gaussian function as is also the case for other pupil bisecting methods [21,22]. Each spot remains compact

<footnote n="1">¹Cambridge Advanced Imaging Centre, University of Cambridge, Downing Site, Cambridge, CB2 3DY, UK</footnote>

<footnote n="2">²Institut de la Vision, 17 rue Moreau, 75012, Paris, France</footnote>

<footnote n="3">³Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK</footnote>

<footnote n="4">⁴School of Physics and Astronomy, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</footnote>

<footnote n="5">⁵School of Food Science and Nutrition, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</footnote>

<footnote>*Corresponding author: ko311@cam.ac.uk</footnote>
throughout the DOF, which is extended with respect to the corresponding widefield microscope since each microlens subtends a small fraction of the full aperture of the microscope objective. Existing optimized algorithms [23,24] can be used to estimate the location of the center of each foci with a precision much finer than its diffraction-limited width. It is well established that information captured within different microlenses can be combined to localize objects in 3D [25,26]. We demonstrate that the temporal sparsity of SMLM, which limits the probability of overlap between diffraction-limited images of distinct emitters, makes it an extremely attractive technique to combine with light field microscopy. We compare two SMLFM configurations tuned to different DOFs by characterizing their 3D precision using fluorescent beads in photon regimes encountered in bioimaging using popular labeling protocols. We demonstrate that SMLFM can accurately resolve features below the diffraction limit using nanorulers with 80 nm separation between binding sites. Finally, efficient detection and localization of single molecules in densely blinking specimens is demonstrated by imaging the membrane of fixed T-cells using both SMLFM configurations, achieving up to 25 3D localizations per frame.

## 2. LIGHT FIELD MICROSCOPY

Light field microscopy (LFM) offers single-shot 3D imaging by simultaneously collecting light from a large, continuous DOF [27]. The emitter location is discriminated through wavefront sampling, generally using a refractive MLA. The MLA partitions a 2D detector into a 2D array of 2D measurements, such that each pixel can be mapped to a 4D space, known as the light field $\mathcal{L}(x, y, u, v)$. Light field measurements encode both the spatial location and arrival direction of incident photons, where $(x, y)$ and $(u, v)$ denote spatial and angular coordinates, respectively. The MLA location in the detection path varies, but it is generally positioned in either a conjugate image plane or a conjugate pupil plane (also known as a back focal or Fourier plane). The MLA location dictates the sampling rates of the spatial and angular coordinates of the light field. When the MLA is placed in an image plane the microlenses themselves sample the spatial domain, generally resulting in a large pixel size. Since this configuration results in a loss of spatial resolution (observed most acutely at the image plane),

a spatially varying 5D PSF, and an axially dependent partition of photons between foci [28], the MLA is optimally located in a plane conjugate to the pupil of the microscope objective for SMLM. This configuration is known as Fourier light field microscopy (FLFM) [29–32]. FLFM provides more homogeneous resolution and signal-to-noise ratio (SNR) than traditional LFM configurations, albeit at the cost of the field of view (FOV) and DOF; however, as demonstrated in this work, the cellular dimensions required for SMLFM can be readily achieved with off-the-shelf components.

In FLFM, each microlens locally apertures the wavefront and generates a focused image, displaced in the direction of, and at a distance proportional to, the average gradient of the apertured wavefront. Hence, as illustrated in Fig. 1, axially displaced emitters are imaged to different positions in each perspective view [33]. The full optical model used to estimate the 3D emitter position is described in detail in Section 1 of Supplement 1, where Eqs. (1) and (2) are derived (c.f. Equations S1–S8). According to this optical model, which calculates the phase in the Fourier plane due to point source displacements from the focal point, the location of the foci in each subaperture, $(x_{uv}, y_{uv})$, is related to the 3D emitter position $(x_i, y_i, z_i)$ according to

$$ \begin{pmatrix} x_{uv} \\ y_{uv} \end{pmatrix} = \begin{pmatrix} 1 & 0 & u\alpha(u, v) \\ 0 & 1 & v\alpha(u, v) \end{pmatrix} \begin{pmatrix} x_i \\ y_i \\ z_i \end{pmatrix}, \quad (1) $$

where $\alpha(u, v)$ is defined as

$$ \alpha(u, v) = \frac{\text{NA}}{n_s \sqrt{1 - \left(\frac{\text{NA}\rho}{n_s}\right)^2}}, \quad (2) $$

where $\rho^2 = u^2 + v^2 = 1$ at the pupil edge and 0 at the optical axis, $k$ is the free space wavenumber, $n_s$ is the sample refractive index, and NA is the numerical aperture of the microscope objective. $u\alpha(u, v)z_i$ and $v\alpha(u, v)z_i$ describe the disparity (parallax) between images in different perspective views. However, since each microlens bounds a finite area and $\alpha(u, v)$ varies nonlinearly with the pupil position, the disparity between 2D localizations in different perspective views is a function of the average wavefront gradient across a given microlens. Full details of how the average

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** (a) Optical layout of a light field microscope with a microlens array positioned in a conjugate pupil plane. (b) The microlens array samples spatial and angular information from the wavefront, which exhibits asymmetric curvature about the primary image plane. Hence, two emitters located at $(x_i, y_i, z_i)$ (red) and $(x_i, y_i, -z_i)$ (blue) are imaged to different positions in each perspective view. (c) Simulated point spread functions for two different light field microscope configurations, with different magnifications of the back focal plane and hence different effective microlens NA (depth of field).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Summary of the algorithm used to estimate 3D emitter position in SMLFM. (a–b) Images of point emitters are detected and localized by Gaussian fitting using traditional 2D SMLM algorithms. Each localization is indexed by the view it appeared in (illustrated here by different shades of gray). Scale bar in (a) represents 15 µm. (a) (inset) Example of an image of a single molecule in a perspective view. (c) Localizations in different views corresponding to the same emitter are identified by applying the constraints of the optical model and removed from the pool. The process is iterated over until no more 2D localizations fit the optical model (d) The ordinary least-squares solution is calculated to give each 3D localization. (e) The 3D localizations are plotted to yield a super-resolved image.</figcaption></figure>

wavefront gradient is calculated are provided in Supplement 1 (c.f. Eqs. S1–S7).

Given a sufficient number of photons, the center of each foci can be estimated with a precision much finer than its width by fitting a 2D Gaussian profile [34–36]. A convenient feature of SMLFM is that existing algorithms and software packages [24] designed and optimized for traditional 2D SMLM can be applied to raw SMLFM data to yield a set of n localizations {(x<sub>uv</sub>, y<sub>uv</sub>)}. ThunderSTORM was used throughout this work [23]. Given this set of localizations, the 3D position of a point emitter, **x** = (x<sub>i</sub>, y<sub>i</sub>, z<sub>i</sub>), can be estimated as the solution to a linear set of equations of the form **Ax** = **b** (c.f. Equations S8 and S10 in Supplement 1), where **b** is a vector representing the set of 2D localizations for a single emitter and **A** describes the disparity between localizations in different perspective views. Since the 2D localizations are normally distributed, ordinary least squares can be used to solve **Ax** = **b** for the 3D emitter position. The accuracy of the optical model used for light field fitting was not found to significantly affect 3D localization precision, which is remarkably robust due to the circular symmetry of optical models of defocus. Localizations in diametrically opposed microlenses exhibit the same, radially flipped, inaccuracy, which results in errors in the estimated axial position with respect to the ground truth. This is demonstrated in Fig. S1 of Supplement 1 for the particular case of inadequately sampling the wavefront gradient. To ensure accurate axial position estimation in SMLFM, calibration scans should be used to verify results. Increasing the accuracy of the optical models reduces the fit error to values approaching the true precision, as determined by repeated experimental and simulated measurements. The fit error is calculated according to the conventional definition for ordinary least squares and is discussed more comprehensively in Section 1 of Supplement 1 (c.f. Eqs. S12–S15).

In most SMLM experiments, it is necessary to detect several hundred thousand localizations to generate high-resolution datasets and achieve Nyquist sampling of the underlying structure [37]. Hence, any viable 3D SMLM approach must be able to detect and localize multiple emitters in each frame. As demonstrated in Fig. 2(c), in SMLFM it is achieved by using Eqs. (1) and (2) to identify the most-likely subset of all 2D localizations in {(*x*<sub>uv</sub>, *y*<sub>uv</sub>)} that correspond to a single emitter. Briefly, the set of localizations {(*x*<sub>uv</sub>, *y*<sub>uv</sub>)} is ordered by decreasing photon number and increasing radial coordinate. Taking each member of this ordered set as a “seed” localization, candidate localizations in other perspective views (within a permitted disparity range set according to the DOF

of the configuration) are identified and grouped. An ordinary least squares solution is calculated to $Ax = b$, to yield $x = (x_i, y_i, z_i)$ for the largest and, hence most-likely, group of 2D localizations. If the 3D SMLFM localization is successful (in terms of having a lower fit error than the threshold), the corresponding group of 2D localizations are removed from the available pool. The process is repeated until no more localizations can be grouped and fitted.

Due to sample and system aberrations, the phase in the pupil plane cannot be entirely accounted for by point source displacements. Since intensity and angular information is captured in SMLFM, it is possible to directly measure aberrations using the 2D localizations themselves, similar to the method used in [38]. For all experimental data presented in this work, these aberrations were estimated by measuring the average residual disparity across the FOV, for emitters located within 0.5 µm of the focal plane. For each perspective view, the residual disparity was estimated as the vector between the measured 2D localizations and the projected localizations predicted by the forward optical model [Eq. (1)] for the fitted 3D position, (x<sub>i</sub>, y<sub>i</sub>, z<sub>i</sub>). The residual disparity was subsequently subtracted from all localizations and the light field fitting algorithm rerun to recover the 3D position of point sources. The 3D SMLFM fitting procedure is summarized in Fig. 2. For further details regarding aberration correction, along with a summary of the parameters used for both 2D localization and light field fitting, refer to Section 1 and Table S1 in Supplement 1.

### 3. SMLFM OPTICAL DESIGN

A standard widefield microscope can be converted to a Fourier light field microscope by adding two components: a relay lens ($L_3$) and a MLA. $L_3$ is placed in a 4f configuration with the tube lens, $L_2$, which images the back focal (Fourier) plane onto the microlens array (MLA). An sCMOS sensor is located at the focal plane of the MLA. The performance of SMLFM is primarily determined by the MLA properties. Following the well-established guidance in terms of magnification and sampling to achieve optimal 2D localization precision, the pixel size in SMLFM should be approximately equal to the standard deviation of each PSF [35], which necessitates the use of small NA (long focal length) microlenses to achieve a suitable pixel size and FOV to image single cells. The DOF of SMLFM is dictated by the effective numerical aperture of each subaperture, defined as the portion of the objective NA spanned by each microlens. The axial sensitivity of SMLFM also depends on the microlens pitch since the disparity between 2D
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** (a–b) Lateral and axial Cramér–Rao lower bounds for configurations 1 and 2. The localization precision, calculated as the standard deviation of 10 repeated estimates is also plotted. The partially illuminated exterior microlenses are not included in the CRLB or localization precision estimates. (c–d) Comparison of the lateral and axial Cramér–Rao lower bounds for configurations 1 and 2 (calculated using all microlenses) with double helix and tetrapod point spread function engineering approaches for 6000 detected photons.</figcaption></figure>

localizations in different perspective views is proportional to the average gradient of the apertured wavefront. Furthermore, one of the most crucial considerations in SMLFM is the division of emitted photons between perspective views, which affects the 2D localization precision. The SNR depends on the total number of microlenses that partition the pupil. To explore the interplay between the microlens pitch and the performance of SMLFM, two different configurations of the light field microscope were built and tested (hereafter referred to as configuration 1 and configuration 2).

Since the robustness of SMLFM to aberrations reduces the requirement for refractive index matching, an oil-immersion objective lens was used to maximize collection efficiency. The lens used also had a pupil diameter that was easily magnified by the tube lens and off-the-shelf achromatic lenses $L_3 = 75$ mm (configuration 1) and $L_3 = 100$ mm (configuration 2) to a diameter approximately equal to an integer number of microlenses. The same, square lattice MLA (SUSS MicroOptics, 18-00672) with a microlens pitch of 1015 µm and a focal length of 26.1 mm was used in both prototypes. These two configurations have differing numbers of illuminated microlenses and magnification. For precise details of both configurations, refer to Table S1 in Supplement 1. The square lattice of the MLA used in our experiments results in partially illuminated microlenses, as illustrated in Figs. 1 and 4. Since this results in distorted PSFs, all data from these microlenses was excluded from analysis in this proof-of-principle work. As a result, the maximum photon throughput was reduced to maximum values of 65% (configuration 1) and 87% (configuration 2). In future iterations, a hexagonal array of microlenses could be used to provide better tessellation of the pupil and a higher photon throughput or, alternatively, data in the exterior microlenses could be deconvolved prior to the 2D localization step.

To evaluate the performance of the SMLFM localization algorithm in both configurations the Cramér–Rao lower bound (CRLB) [36] was computed for simulated images with an average of 6000 detected photons. Details of the simulations used to generate the data are presented in Section 2 of Supplement 1.

Figure 3 demonstrates that both configurations exhibit a theoretical precision below 20 nm throughout a 5 µm DOF. Furthermore, the precision achieved using the SMLFM localization algorithm outlined in Section 2 approaches the CRLB at the focal plane. While the CRLB does not predict isotropic precision, the axial and lateral localization precision estimates exhibit a close relationship. Discrepancies between the achieved localization and the CRLB increase with axial distance from the focal plane, but lateral and axial precision remain below 20 nm and 40 nm, respectively, throughout the axial range. It is primarily due to the fact that, for the same number of photons detected at the sensor, the average number of photons per light field fit decreases with an increasing axial emitter position. In this exploration of the feasibility of SMLFM, 3D point position estimation is based solely on the geometry of the constellation of 2D localizations in different perspective views for each emitter. It is anticipated that improvements in localization precision could be achieved by extending the optical model beyond the current analysis to include a greater number of parameters. In particular, changes in the lateral profile of each foci with an axial position could be incorporated. Furthermore, different perspective views could be weighted according to their information content and the simple least-squares fitting algorithm could be replaced with a maximum likelihood approach.

As expected, the two SMLFM configurations differ in terms of maximum precision and DOF. Configuration 1, having fewer lenses (and therefore a larger effective NA), achieves a slightly better lateral precision at the focal plane at the expense of DOF.
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** (a) Images of fluorescent beads in different views for configuration 1 and configuration 2. The red circles illustrate the wavefront diameter in the BFP. The white lines indicate the microlens edges. In both configurations, the corner microlenses were partially illuminated and were not used for localization. Scale bars represent 1 µm. (b–c) Lateral and axial localization precision (circular markers) and fit error ($Δ_x, Δ_z$) as a function of the axial position (z) of an emitter for (b) configuration 1 and (c) configuration 2. (d) $1/\sqrt{N}$ fit of lateral and axial precision is plotted as a function of number of photons for configurations 1 and 2. Circular markers represent the average value of precision (for a bin width of 500 photons). The full dataset is plotted in Fig. S8 of Supplement 1. (e–f) 50 nm steps can be resolved using both configurations. (g–i) Histograms (12 nm bin width) of single molecule localizations for three representative nanorulers. Gaussian fits are also plotted, with estimates of average distance between emitters indicated for each nanoruler. (h) Inset: Image of a single nanoruler (scale bar represents 25 nm), $σ_x$ corresponds to the average standard deviation of the Gaussian fit.</figcaption></figure>

However, both configurations achieve sub-5 nm lateral precision at the focal plane. The similarity of the curves plotted in Figs. 4 (a) and (b) suggest that the configuration with the largest DOF is preferable. A caveat is that decreasing the photon flux per microlens increases the noise floor below which 2D detection and localization become difficult (refer to Fig. S8 and S9 in Supplement 1). As such, the choice of MLA in SMLFM must take into account the achievable SNR with the depth of field required.

At photon fluxes routinely achievable using common labeling protocols for single molecule imaging, the theoretical performance of SMLFM is comparable to leading 3D SMLM techniques such as double helix and saddle point (tetrapod), as summarized in Figs. 3(c) and 3(d). Furthermore, detection and localization procedures used in double helix and saddle point SMLM are generally more complex than the 2D localization and least-squares light field fitting algorithm implemented for SMLFM since additional steps such as template matching are necessary. 3D localization in SMLFM is as straightforward as astigmatism, which is one of the most widely used 3D SMLM approaches due to its relative simplicity. The ease of implementation of in situ aberration correction in SMLFM is another key advantage with respect to other 3D SMLM techniques, where it is generally necessary to estimate the experimental pupil function (for instance, using phase retrieval)

and incorporate it into the optical model to improve experimental localization precision.

## 4. RESULTS AND DISCUSSION

To benchmark the performance of SMLFM, a 2D sample comprised of 100 nm fluorescent beads (TetraSpeck Fluorescent Microspheres Kit (T14792), ThermoFisher) immobilized on a coverslip (#1.5 thickness) was imaged. Data was acquired as the microscope stage was translated in fixed steps of 50 nm along the optical axis. For both configurations, 4000 net photons were detected on average across the entire axial range. The localization precision was calculated as the standard deviation of the fitted 3D position at each 50 nm step with 10 repeats at an exposure of 10 ms. Fig. 4 presents a summary of results. For configuration 1, isotropic lateral and axial localization precision was measured, remaining below 20 nm throughout a 3 µm imaging depth (below 50 nm over a 4 µm axial range). Similar to the simulations, configuration 2 exhibited a larger DOF with the isotropic lateral and axial precision remaining below 20 nm over an extended 5 µm range. At an average photon flux of 4000 per event, this value is competitive with other 3D localization techniques [11,12,39]. The data presented in Figs. 4(b) and 4(c), demonstrate that the fit error is a robust
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** (a) Background subtracted camera frame of Alexa-647 captured using SMLFM (configuration 1). Scale bar represents 15 µm. (b) Insets: Images captured in each perspective view. (c) Histogram showing the distribution of the number of photons emitted per event. A median of 5569 photons were collected per event. (d) Photobleaching curves of Alexa-647 imaged with light field microscopy. The normalized, integrated intensity in each perspective view is plotted as a function of time. Traces from different perspective views are distinguished by color (Visualization 1 and Visualization 2).</figcaption></figure>

upper bound for the precision across all depths and can be used to evaluate the quality of each fit.

The relationship between SMLFM localization precision and the number of photons was measured by varying the laser power to explore a range of net detected photons. The 10 ms exposure time was kept constant. This data was acquired over an axial range of 4 µm (configuration 1) and 7 µm (configuration 2). The localization precision was again calculated as the standard deviation of the fitted 3D position across 20 repeats (10 repeats for configuration 2). A summary of results is presented in Fig. 4(d), a $1/\sqrt{N}$ fit of lateral and axial precision is plotted as a function of the number of photons for configurations 1 and 2. Circular markers representing the average value of precision (bin width of 500 photons) have also been plotted. The full dataset is plotted in Fig. S8 of Supplement 1. At low photon numbers, configuration 1 exhibits better performance than configuration 2, due to the higher number of photons per microlens, resulting in higher SNR and a better 2D localization precision. The $1/\sqrt{N}$ fit does not capture the behavior exhibited by configuration 2 at low photon numbers where 2D detection and localization become difficult due to the higher noise floor reached at around 1000 photons. However, at sufficiently high photon numbers, the performances of the two configurations become comparable. The precision floor of both configurations is approximately isotropic with values of 8 nm (configuration 1) and 10 nm (configuration 2), respectively. Furthermore, a linear, monotonic relationship between the fit and stage position was observed in the case of both configuration 1 and configuration 2, as shown in Figs. 4(e) and 4(f). Crucially, clear contrast can be observed in the 50 nm axial steps. To conclusively demonstrate that SMLFM can accurately resolve features below the diffraction limit, DNA origami nanorulers (GATTA-PAINT 80 nm, GATTAQuant GmbH) were imaged. 1D histograms of

three representative nanorulers are plotted in Figs. 4(g)–4(h), which show an average spacing of 82.8 ± 1.2 nm and a standard deviation of 21.4 ± 0.3 nm. Altogether, the experimental results presented in Fig. 4 confirm the viability of single molecule light field microscopy.

The single molecule sensitivity of SMLFM was conclusively demonstrated by imaging Alexa-647 dispersed on a coverslip (#1.5 thickness) with a 70 ms exposure time. Fluorescent traces exhibiting discrete signal levels, characteristic of single molecule photobleaching events were observed [40]. Figure 5 shows an example of single step photobleaching of a fluorophore at the tail end of the distribution of typical localized molecules acquired in this experiment. Traces of the integrated intensity from images of the emitter in each perspective view are plotted. As expected, spatio-temporal correlations are observed between the measurements from different views.

To examine the superresolution structural imaging capabilities of SMLFM, we imaged the membrane of fixed Jurkat T-cells using point accumulation for imaging of nanoscale topography (PAINT), based on the stochastic binding of fluorescent wheat germ agglutinin [41]. Cells were imaged using a HILO illumination to reduce background. Datasets comprised of 45,000 to 150,000 images were acquired over 1 to 3 h. Refer to Supplement 1 for full details of the experimental parameters. Typical frames, which capture information throughout the depth of field, contained 45 2D localizations, corresponding to, on average, 13 3D light field localizations. After filtering by fit error (using an upper limit of 80 nm) experiments achieved averages between three and nine light field localizations per frame. The density of localizations practically achieved in our SMLFM experiments is ≈600,000 localizations (≥300,000 post-filtering) over 45,000 frames with a FOV of 15 µm × 15 µm. This high density is enabled by the
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Super-resolved images of Jurkat T cells captured with single molecule light field microscopy. (a) (i) Horizontal and vertical (ii) cross-sections through data acquired with configuration 1. Scale bar represents 2 µm. Inset: Line profiles plotted for *xy* and *zy* projections through a microvilli demonstrate that features can resolved beyond the diffraction limit in SMLFM. (b) Kernel density plots summarizing the characteristics of the acquired data. Data with fit error below 80 nm, indicated by the red dashed line in (ii) was filtered into the final visualizations. (i) Number of photons per light field fit for the filtered localizations. (ii) 3D fit error and axial position for all light field localizations. (iii) Axial locations of filtered localizations. (c) (i) Horizontal and vertical (ii) cross-sections through data acquired with configuration 2.</figcaption></figure>

ability to fit multiple molecules with the same lateral position but differing axial position and the relatively small extent of the SMLFM PSF. This localization density is competitive with other large DOF 3D SMLM approaches that have been practically demonstrated on whole cells to generate data over a DOF greater than 2 µm. For instance, saddle point, double helix PSF and MPM have been used successfully in such experiments to generate up to 33 total localizations per frame and up to three filtered localizations per frame [8, 11, 17]. In comparison, our results demonstrate 13 total light field localizations per frame and nine filtered light field localizations per frame. Visualizations of these filtered SMLFM localizations are shown in Fig. 6 for two such experiments, one for each SMLFM configuration. Horizontal and vertical projections through each cell demonstrate that the resolution of SMLFM is sufficient to resolve the 3D membrane contour and microvilli.

## 5. CONCLUSION

We have demonstrated the viability of SMLFM for scanless 3D superresolution imaging. Our results show that SMLFM can localize single molecules with a near-isotropic precision of 20 nm using only a few thousand emitted photons, a comparable performance to other 3D imaging techniques [3–6]. We have also demonstrated detection and 3D localization of single molecules in densely blinking specimens, achieving up to 25 light field localizations per frame in data sets of 40,000 to 150,000 frames. The mechanism that enables SMLFM, disparity between perspective views, is one that reveals the underlying wavefront structure and amplitude of the field in the pupil. Such data enables post-acquisition aberration correction without requiring phase retrieval or z-dependent calibration scans. This rich information coupled with the simple PSF

footprint and the optical properties of MLAs result in SMLFM having the potential to offer highly accurate and precise multicolor 3D nanoscopy over whole eukaryotic cell volumes.

**Funding.** Engineering and Physical Sciences Research Council (EP/G037256/1, EP/L015455/1, EP/M003663/1, EP/R025398/1); Cambridge Commonwealth, European and International Trust; Higher Education Commission, Pakistan; Wellcome Trust (212936/Z/18/Z); Royal Society (RGF\EA\181021, uf120277, URF\R\180029).

**Acknowledgment.** The authors would like to thank Dr. Michael Shaw for useful discussions.

Kevin O’Holleran conceptualized the project. O’Holleran and Steven F. Lee supervised and administered the project. Ruth R. Sims, Sohaib Abdul Rehman, Leila Muresan, and O’Holleran developed the methodology. Sims, Abdul Rehman, Adam Clark, Martin O. Lenz, and O’Holleran conducted the experimental investigation. Edward W. Sanders and Aleks Ponjavic provided the cell samples and labeling methodology. Sims, Abdul Rehman, Sarah I. Benaissa, Ezra Bruggeman, Muresan, and O’Holleran performed formal analysis of data and developed software. Sims was responsible for visualization of results in the paper. Sims, Abdul Rehman, and O’Holleran wrote the paper. All authors reviewed and edited the final paper.

**Disclosures.** The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

†These authors contributed equally to this paper.
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Single molecule light field microscopy:
supplement

RUTH R. SIMS,<fnref n="1" />,<fnref n="2" />,† SOHAIB ABDUL REHMAN,<fnref n="1" />,† ID MARTIN O. LENZ,<fnref n="1" /> ID SARAH I. BENAISSA,<fnref n="1" /> EZRA BRUGGEMAN,<fnref n="3" /> ID ADAM CLARK,<fnref n="3" /> EDWARD W. SANDERS,<fnref n="3" /> ID ALEKS PONJAVIC,<fnref n="3" />,<fnref n="4" />,<fnref n="5" /> LEILA MURESAN,<fnref n="1" /> STEVEN F. LEE,<fnref n="3" /> AND KEVIN O'HOLLERAN<fnref n="1" />,\* ID

This supplement published with The Optical Society on 21 August 2020 by The Authors under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/) in the format provided by the authors and unedited. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Supplement DOI: https://doi.org/10.6084/m9 figshare.12824225

Parent Article DOI: https://doi.org/10.1364/OPTICA.397172

<footnote n="1">¹Cambridge Advanced Imaging Centre, University of Cambridge, Downing Site, Cambridge, CB2 3DY, UK</footnote>

<footnote n="2">²Institut de la Vision, 17 rue Moreau, 75012, Paris, France</footnote>

<footnote n="3">³Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK</footnote>

<footnote n="4">⁴School of Physics and Astronomy, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</footnote>

<footnote n="5">⁵School of Food Science and Nutrition, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</footnote>

<footnote>†These authors contributed equally to this paper.</footnote>

<footnote>*Corresponding author: ko311@cam.ac.uk</footnote>
# Single Molecule Light Field Microscopy: supplementary material

RUTH R. SIMS<fnref n="1" />,<fnref n="2" />,⁺, SOHAIB ABDUL REHMAN<fnref n="1" />,⁺, MARTIN O. LENZ<fnref n="1" />, SARAH I. BENAISSA<fnref n="1" />, EZRA BRUGGEMAN<fnref n="3" />, ADAM CLARK<fnref n="3" />, EDWARD W. SANDERS<fnref n="3" />, ALEKS PONJAVIC<fnref n="3" />,<fnref n="4" />,<fnref n="5" />, LEILA MURESAN<fnref n="1" />, STEVEN F. LEE<fnref n="3" />, AND KEVIN O'HOLLERAN<fnref n="1" />

Compiled August 18, 2020

This document provides supplementary information to “Single Molecule Light Field Microscopy.”

## 1. 3D POINT SOURCE LOCALIZATION IN SINGLE MOLECULE LIGHT FIELD MICROSCOPY

Ideally, a microscope objective transforms a spherical wave emitted by a point source located at the origin $(x_p, y_p, z_p)$ into a plane wave at the pupil. Spherical waves emitted by a point emitter displaced from the origin, located at $(x_i, y_i, z_i)$, generate more complex wavefronts in the pupil since in this case the optical path length to the principal sphere varies as a function of propagation direction. The majority of microscope objectives are designed according to the sine condition, according to which, ray height is preserved during mapping between principal planes $(\{x_p, y_p\} \to \{u, v\})$ [11]. As a result, the phase difference in the pupil between wavefronts emanating from points located at $(x_p, y_p, z_p)$ and $(x_i, y_i, z_i)$ depends on the optical path difference (OPD) between two rays that intersect at the principal sphere:

$$ \begin{aligned} \text{OPD} &= n_s (|\vec{d}| - f) = n_s (|\vec{f} - \vec{r}| - f) \\ &= n_s \sqrt{(x_p - x_i)^2 + (y_p - y_i)^2 + (\sqrt{f^2 - x_p^2 - y_p^2} - z_i)^2} - n_s f \end{aligned} \quad (S1) $$

For small displacements from the focal point, the quotient $(x_i^2 + y_i^2 + z_i^2) / f^2$ vanishes and Equation S1 takes the form:

$$ \text{OPD} \sim \frac{n_s}{f} \left[ (x_i x_p + y_i y_p) + z_i \sqrt{f^2 - x_p^2 - y_p^2} \right] \quad (S2) $$

Converting OPD to phase by multiplying by the wavenumber $k = 2\pi/\lambda_0$, where $\lambda_0$ is the free space wavelength, the phase in the pupil plane resulting from point source displacements from the focal point is:

$$ \Phi_{x_i, y_i, z_i} = \frac{n_s k}{f} \left[ (x_i u + y_i v) + z_i \sqrt{f^2 - u^2 - v^2} \right] \quad (S3) $$

In normalized pupil co-ordinates, where $\rho^2 = u^2 + v^2 = 1$ at the pupil edge and 0 at the optical axis, Equation S3 becomes:

$$ \Phi_{x_i, y_i, z_i} = kNA (x_i u + y_i v) + n_s z_i k \sqrt{1 - \left(\frac{NA\rho}{n_s}\right)^2} \quad (S4) $$

In Fourier light field microscopy the phase in the pupil plane is sampled by a microlens array. Each microlens generates a focused image displaced in the direction of, and, at a distance proportional to, the average phase gradient across each microlens. The centre of each image can be estimated by fitting a 2D Gaussian profile using algorithms designed and optimized for traditional 2D single molecule localization microscopy to yield a set of 2D localizations $\mathbf{b} = \{(x_{uv}, y_{uv})\}$.

<footnote n="1">¹Cambridge Advanced Imaging Centre, University of Cambridge, Downing Site, Cambridge, CB2 3DY, UK</footnote>

<footnote n="2">²Institut de la Vision, 17 rue Moreau, 75012, Paris, France</footnote>

<footnote n="3">³Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK</footnote>

<footnote n="4">⁴School of Physics and Astronomy, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</footnote>

<footnote n="5">⁵School of Food Science and Nutrition, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UK</footnote>

<footnote>⁺These authors contributed equally to this work</footnote>
The gradient of the phase in the pupil resulting from point source displacements is:

$$ \nabla \Phi = kNA \begin{pmatrix} x_i + u\alpha(u,v)z_i \\ y_i + v\alpha(u,v)z_i \end{pmatrix} \qquad (S5) $$

such that $u\alpha(u,v)z_i$ and $v\alpha(u,v)z_i$ describe the disparity (parallax) between localizations in different perspective views, where $\alpha(u,v)$ is defined:

$$ \alpha(u, v) = \frac{-NA}{n_s \sqrt{1 - \left(\frac{NA\rho}{n_s}\right)^2}} \qquad (S6) $$

and each 2D localization, $(x_{uv}, y_{uv})$, is related to 3D point source position, $\mathbf{x} = (x_i, y_i, z_i)$; according to:

$$ \begin{pmatrix} x_{uv} \\ y_{uv} \end{pmatrix} = \begin{pmatrix} 1 & 0 & u\alpha(u, v) \\ 0 & 1 & v\alpha(u, v) \end{pmatrix} \begin{pmatrix} x_i \\ y_i \\ z_i \end{pmatrix} \qquad (S7) $$

Note that a factor $1/kNA$ factor has been included to convert co-ordinates from image space to sample space.

Hence, given a set of 2D localizations from $n \ge 2$ distinct perspective views, $\mathbf{b} = \{(x_{uv}, y_{uv})\}$, the 3D location of an point source can be estimated as the solution to:

$$ \begin{pmatrix} 1 & 0 & u_1\alpha(u_1, v_1) \\ \vdots & \vdots & \vdots \\ 1 & 0 & u_n\alpha(u_n, v_n) \\ 0 & 1 & v_1\alpha(u_1, v_1) \\ \vdots & \vdots & \vdots \\ 0 & 1 & v_n\alpha(u_n, v_n) \end{pmatrix} \begin{pmatrix} x_i \\ y_i \\ z_i \end{pmatrix} = \begin{pmatrix} x_1 \\ \vdots \\ x_n \\ y_1 \\ \vdots \\ y_n \end{pmatrix} \qquad (S8) $$

For both configurations presented in this work, the square microlenses which partition the pupil bound a finite area, $A = d^2$ for microlens pitch $d$. Hence it is necessary to calculate the average wavefront gradient over the pupil co-ordinates spanned by each microlens according to:

$$ \overline{\nabla \Phi}_k = \begin{pmatrix} \int_{p_k} \nabla \Phi du \\ \int_{p_k} \nabla \Phi dv \end{pmatrix} = \begin{pmatrix} x_i + \frac{n_s z_i}{dNA} \left( \sqrt{1 - \frac{NA^2}{n_s^2} \left( \left[ u_k + \frac{d}{2} \right]^2 + v_k^2 \right)} - \sqrt{1 - \frac{NA^2}{n_s^2} \left( \left[ u_k - \frac{d}{2} \right]^2 + v_k^2 \right)} \right) \\ y_i + \frac{n_s z_i}{dNA} \left( \sqrt{1 - \frac{NA^2}{n_s^2} \left( u_k^2 + \left[ v_k + \frac{d}{2} \right]^2 \right)} - \sqrt{1 - \frac{NA^2}{n_s^2} \left( u_k^2 + \left[ v_k - \frac{d}{2} \right]^2 \right)} \right) \end{pmatrix} \qquad (S9) $$

where $(u_k, v_k)$ are the co-ordinates of the centre of the $k$th microlens. In this case Equation S8 takes the form:

$$ \begin{pmatrix} 1 & 0 & \xi_u(u_1, v_1) \\ \vdots & \vdots & \vdots \\ 1 & 0 & \xi_u(u_n, v_n) \\ 0 & 1 & \xi_v(u_1, v_1) \\ \vdots & \vdots & \vdots \\ 0 & 1 & \xi_v(u_n, v_n) \end{pmatrix} \begin{pmatrix} x_i \\ y_i \\ z_i \end{pmatrix} = \begin{pmatrix} x_1 \\ \vdots \\ x_n \\ y_1 \\ \vdots \\ y_n \end{pmatrix} \qquad (S10) $$

where $\xi_u(u_k, v_k)$ and $\xi_v(u_k, v_k)$ are defined:

$$ \begin{align} \xi_u(u_k, v_k) &= \frac{n_s}{dNA} \left( \sqrt{1 - \frac{NA^2}{n_s^2} \left( \left[ u_k + \frac{d}{2} \right]^2 + v_k^2 \right)} - \sqrt{1 - \frac{NA^2}{n_s^2} \left( \left[ u_k - \frac{d}{2} \right]^2 + v_k^2 \right)} \right) \nonumber \\ \xi_v(u_k, v_k) &= \frac{n_s}{dNA} \left( \sqrt{1 - \frac{NA^2}{n_s^2} \left( u_k^2 + \left[ v_k + \frac{d}{2} \right]^2 \right)} - \sqrt{1 - \frac{NA^2}{n_s^2} \left( u_k^2 + \left[ v_k - \frac{d}{2} \right]^2 \right)} \right) \tag{S11} \end{align} $$
Equation S10 was solved using Equation S11 for all data acquired where NA ≥ n<sub>s</sub>, however in the case of an aqueous sample medium, ξ<sub>v</sub>(u<sub>k</sub>, v<sub>k</sub>) and ξ<sub>v</sub>(u<sub>k</sub>, v<sub>k</sub>) are complex for some pupil co-ordinates bounded by the exterior microlenses (for more details refer to Section 3). In this case, the average phase gradient was calculated numerically using 20<sup>2</sup> points spanning each microlens as illustrated in (E) of Figure S1. This approach was validated by checking that in the case of NA = n<sub>s</sub>, the discrete estimate converged to the analytical value.

<figure><img src="image_9.png" /><figcaption>**Fig. S1.** (A) Spherical phase Φ<sub>x,y,z</sub> in the pupil due to axial displacement of a point source from the origin. (B) Derivative of the phase in (A), ∇Φ<sub>x,y,z</sub>. (C-D) ∇Φ plotted as a function of number of pupil samples for a central row of microlenses. Dashed black lines indicate microlens edges. 1 sample corresponds to the microlens centre. Each individual line is transparent, hence opaque regions indicate convergence of ∇Φ. According to these graphs, using an insufficient number of pupil-coordinates to calculate ∇Φ leads to an underestimation of axial emitter position (which is proportional to average phase gradient). This error increases with axial distance from the focal plane. (E) Since each microlens spans a finite area, and ∇Φ<sub>x,y,z</sub> varies non-linearly as a function of pupil position, it is necessary to calculate ∇Φ. (F) ∇Φ is plotted against the number of pupil-coordinate samples used to calculate the average for configurations 1 (opaque, continuous lines) and 2 (transparent, dashed lines). Values were calculated for the microlens highlighted in (E) and the microlens in teh corresponding position for configuration 2. In the cases where Equation S11 could not be evaluated analytically, for instance due to complex pupil co-ordinates, a meshgrid of 20 equidistant pupil co-ordinate samples within the each microlens was found to be sufficient for both configurations.</figcaption></figure>

There are many different approaches to solving equations of the form Ax = b (Equations S8 and S10). In this study of the feasibility of SMLFM in low photon regimes, the ordinary least-squares solution to the equation Ax = b was calculated according to:

$$
\mathbf{x} = (A^T A)^{-1} A^T \mathbf{b} \qquad (S12)
$$

with mean squared error (MSE):

$$
MSE = \frac{1}{n-m} (\mathbf{b}^T \mathbf{b} - \mathbf{b}^T A \mathbf{x}) \qquad (S13)
$$
where $n \ge 2$ corresponds to the number of 2D localizations from different perspective views used in each light field localization and $m = 3$ corresponds to the 3 spatial dimensions. Finally the standard error in each localization is:

$$ (\Delta x_i, \Delta y_i, \Delta z_i) = \sqrt{\mathrm{diag} \left[ (A^T A)^{-1} \mathrm{MSE} \right]} \qquad (S14) $$

which was used to calculate the 3D error of each light field fit ($\Delta_{3D}$) according to:

$$ \Delta_{3D} = \sqrt{\Delta x_i^2 + \Delta y_i^2 + \Delta z_i^2} \qquad (S15) $$

The accuracy of the least-squares solution to Equation S10 was evaluated by comparing the average fit error to the standard deviation of location estimates of repeat measurements on multiple images of beads imaged at low laser power. The results are presented in Figure S2 and demonstrate that, for both configurations, the fit error is an upper bound for the precision of the estimation of the emitter location. In the SMLFM localization approach outlined above, 3D point position estimation is based solely on the geometry of the constellation of 2D localizations in different perspective views for each emitter; the effect of the point source position on the phase in the Fourier plane has been accounted for but polarization and intensity effects have not been considered. Since the fit error is a measure of the geometric deviation between the measured localizations and the projected localizations from the fitted $(x_i, y_i, z_i)$, we used the 3D fit error rather than the inferred precision based on the total number of photons per light field fit to assess the quality of each light field fit.

<figure><img src="image_10.png" /><figcaption>**Fig. S2.** Histograms demonstrating that the fit error is an upper bound for the precision of the estimation of the location of the emitter for data acquired in both configurations. Dashed line indicates 1:1 correspondence between fit error and precision. Histogram bin size 1.5 nm. Colour bar valid for all plots.</figcaption></figure>

In different SMLFM configurations, the average phase gradient is calculated over different portions of the pupil. The effect of averaging the phase gradient across each microlens is demonstrated in Figure S1 where $\Phi$ and $\nabla\Phi$ are plotted as a function of axial displacement. (C) and (D) of the same figure demonstrate that in the case of both configurations 1 and 2, undersampling the pupil co-ordinates (for instance by taking the central co-ordinate of each microlens), leads to an inaccurate estimate of the axial displacement of a point source from the origin. The error is largest for the exterior microlenses corresponding to the maximum curvature of $\Phi_{x,y,z}$. As the number of
pupil samples is increased, $\nabla\Phi$ converges to Equation S9 and, ultimately, the correct value of axial position.

When imaging multiple point sources, the subset of 2D localizations corresponding to each emitter must be identified before calculating the least-squares solution to Equation S10. This was achieved by iterating over the localizations in a perspective view proximal to the optical axis. For each seed localization, candidate localizations in other perspective views (within a permitted disparity range set according to the depth of field of the configuration) were identified. If a sufficient number were found, the 3D position of the emitter was calculated as outlined. If $\Delta_{3D}$ was below a pre-defined threshold, the light field fit was approved and all corresponding 2D localizations removed from the pool. This procedure was continued until no more light field fits with 3D fit error below threshold could be calculated.

<figure><img src="image_11.png" /><figcaption>**Fig. S3.** Summary of filtering of localizations into final datasets as rendered in Figure 6 of the main text. Top row (configuration 1, (A) of Figure 6), bottom row (configuration 2, (C) of Fig 5). 2D localizations per frame refers to all localizations in all views returned by the ThunderSTORM software [15]. The light field fitting algorithm returns 3D position estimates referred to as LF localizations per frame. These 3D localizations are filtered using 3D fit error as a proxy for precision to yield so-called LF localizations per frame (err < 80 nm). The final histogram in each row shows the number of views (i.e. number of 2D localizations) which contributed to each light field fit.</figcaption></figure>

The final aspect of the 3D localization algorithm is the estimation and correction of sample and system aberrations. Wavefront aberrations also contribute to the phase in the pupil and hence are manifest in SMLFM as displacements (termed residual disparity) of the 2D localizations $(x_{uv}, y_{uv})$ from the localizations predicted by the optical model for the fitted localization. For all data presented in this work, an estimate of the global aberrations was calculated as the average residual disparity in each perspective view, for point sources located within 0.5 µm of the focal plane. The residual disparity was calculated as the error in each $(x_{uv}, y_{uv})$ after projecting the fitted $(x_i, y_i, z_i)$ solution into each perspective view. This results in a vector field across $(u, v)$ which can be averaged over a large number of light field localizations. The stochastic error in the residual vector error cancels out when a sufficient number of localizations are averaged. Any residual vector is a systematic error corresponding to a deviation in the wavefront relative to our optical model - an aberration. The residual disparity was subtracted from all 2D localizations and the least-squares solution to Equation S10 was re-calculated to recover the 3D position of point sources.

The axial extent of the isoplanatic patch was investigated by observing the average fit error as a function of z position. Light field localizations, for both the 'bead scan' and 'T-cell' datasets, were partitioned into groups according to axial location. The average fit error was calculated for each 1 µm segment throughout the depth of field. Figure S4 demonstrates that for all datasets and for both configurations, subtracting the residual disparity measured within 0.5 µm of the focal plane reduced the fit error within a 4 µm depth of field. The axial dependency of the fit error in the case of the bead scan is due to spherical aberration resulting from using an oil objective to image into an aqueous medium - which changes as a function of axial separation between the 2D bead layer and the microscope objective as discussed in Section 3.
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

**Fig. S4.** Subtracting the residual disparity from all 2D localizations reduces 3D fit error throughout the depth of field for all datasets presented in this work.

<figure><img src="image_16.png" /></figure>

**Fig. S5.** Average 3D fit error plotted for each perspective view as a function of axial emitter position. In each view the coloured and grey lines represent the average fit error for two opposing halves of the field of view.

To test the lateral variance of the fit error for the raw data, the field of view was divided horizontally into two regions. The average fit error was calculated for each region and results are shown in Figure S5, which demonstrates that $\Delta_{3D}$ exhibits the same profile for both sides of the FOV, with some divergence near the edge of the imaging volume. As demonstrated in Figure S6, for all datasets presented in this work, subtraction of the residual disparity improves the quality of the fit. In principle, spatially varying sam-
ple dependent aberrations could be corrected for in SMLFM, though exploring this further was beyond the scope of this feasibility study.

<figure><img src="image_17.png" /><figcaption>**Fig. S6.** Subtracting residual disparity from all 2D localizations reduces 3D fit error. Insets show the magnitude of the residual disparity for each dataset. Lateral fit error is plotted on the left side of each graph and axial fit error on the right hand side as indicated. Colour bar valid for all plots.</figcaption></figure>
The entire procedure used to estimate 3D emitter location in SMLFM is outlined below:

1. Single molecule detection and 2D Gaussian fitting on raw data to yield a set of 2D localizations $\{(x_{pix}, y_{pix})\}$

2. Transform from camera to light field co-ordinates in the pupil plane $\{(x_{pix}, y_{pix})\} \rightarrow \{(x_{uv}, y_{uv})\}$ by translation and scaling

3. Filter $\{(x_{uv}, y_{uv})\}$ according to the standard deviation of each 2D Gaussian fit. Remove localizations in partially illuminated exterior lenses

4. Estimate $\theta_{MLA}$ as that which maximizes number of fitted light field points with minimum fit error, rotate $\{(u, v)\}$ by $-\theta_{MLA}$

5. Estimate system and sample aberrations as the residual disparity of points located within 0.5 µm of the focal plane. Subtract the residual disparity from all 2D localizations

6. Order the $q$ localizations in a view proximal to the optical axis by decreasing number of photons. Loop over each of the $q$ localizations:

(a) Identify candidate localizations in other views within a permitted disparity range according to the optical model

(b) In the case of multiple candidates within a perspective view, choose the localization which minimizes the fit error

(c) If corresponding localizations are found in a minimum number of perspective views, solve Equation S10

(d) If the fit error of the solution is below threshold remove these localizations from the set $\{(x_{uv}, y_{uv})\}$

A summary of the parameters and values found to optimize fitting in SMLFM is presented in Table S1. Whilst this procedure is generally applicable to all SMLFM data, optimization of the parameters in Table S1 would be necessary for data acquired on systems substantially different than those described in this work.

<table><caption>**Table S1.** Summary of fit parameters used for 3D position estimation in SMLFM.</caption><thead><tr><th>Parameter (units)</th><th>Configuration 1</th><th>Configuration 2</th></tr></thead><tbody><tr><td>2D localizations per emitter (max)</td><td>5</td><td>12</td></tr><tr><td>σ<sub>min</sub> for 2D fits (pixels)</td><td>1.1</td><td>1.1</td></tr><tr><td>σ<sub>max</sub> for 2D fits (pixels)</td><td>1.8</td><td>1.8</td></tr><tr><td>θ<sub>search</sub> for candidates (°)</td><td>1</td><td>1</td></tr><tr><td>Permitted disparity range (pixels)</td><td>0.3</td><td>0.3</td></tr><tr><td>Minimum number of views for LF fit, N<sub>min</sub></td><td>2</td><td>2</td></tr><tr><td>3D Fit threshold (μm)</td><td>1</td><td>1</td></tr><tr><td>Range for residual disparity calculation (μm)</td><td>1</td><td>1</td></tr></tbody></table>

The square lattice of the MLA used in our experiments resulted in partially illuminated microlenses with distorted PSFs which could not reasonably be approximated by a 2D Gaussian function. Masking the exterior microlenses impacts configuration 1 to a greater extent than configuration 2 by reducing the photon throughput to 66 and 87 percent respectively (this can be seen by examining the CRLB plots in Figure 3 of the main text). Exclusion of these exterior microlenses was found to affect the axial localization precision much more than the lateral localization precision. This can be understood by considering the phase in the back focal plane due to lateral and axial point source displacements. According to Equation S4, lateral point source displacements result in a tilted phase in the pupil with constant derivative. Consequently, neglecting intensity variations, information about the lateral position of a point source is equally distributed between perspective views. On the other hand, two regimes of $\nabla\Phi_z$ are apparent. For small values of $\rho$, where the paraxial (small-angle) approximation is valid, the wavefront due to defocused point sources is a quadratic function of pupil co-ordinates such that $\nabla\Phi$ varies linearly with $u$ and $v$, whereas, at the edge of the pupil, $\nabla\Phi$ increases sharply and is no longer a linear function of $u$ and $v$. This suggests that in Fourier SMLFM, the exterior microlenses are more sensitive to axial emitter location than those located closer to the optical axis. On this basis, it would appear optimal to utilize microlenses with small pitch at the edge of the pupil in order to maximize the light field point spread function PSF sensitivity to axial position. However, this does not take into account the fact that smaller microlenses collect fewer photons, resulting in poorer 2D localization precision. Furthermore, since the rate of increase of wavefront curvature is fastest for the peripheral microlenses, the microlenses most sensitive to axial emitter location, exhibit the lowest depths of field. This is demonstrated in Figure S7 where the average width of the 2D Gaussian fit to an image of a point emitter is plotted as a function of axial emitter position. Identifying the optimal partition of the pupil for SMLFM is beyond the scope of this proof-of-principle work.

Note that in this approach, 3D point source localization is based solely on the geometry of the constellation of 2D localizations in different perspective views for each emitter; the effect of the point source position on the phase in the Fourier plane has been accounted for but polarization and intensity effects are not considered. Furthermore, each perspective view is intrinsically assumed to contribute equal information, which is not necessarily the case due to the non-linearity of disparity (c.f. Equation S6). We anticipate that extending the optical model to include these effects and using more complex fitting approaches such as weighted least-squares, maximum likelihood or compressed sensing would improve localization precision.
<figure><img src="image_18.png" /><figcaption>**Fig. S7.** Average width of the 2D localizations in each perspective view as a function of z. (A) Configuration 1, (B) Configuration 2. The asymmetry in beam width as a function of axial position clearly present in both cases between the principal axes indicates the presence of astigmatism.</figcaption></figure>

The relationship between localization precision and number of photons was measured by varying the laser power to explore a range of net detected photons up to 10,000. The localization precision was again calculated as the standard deviation of the fitted 3D position across 20 repeats (10 repeats for configuration 2). Data plotted in Figures S8 and S9 demonstrates that the precision floor of both configurations is approximately isotropic with values of 8 nm (configuration 1) and 10 nm respectively (configuration 2). There are discrepancies between the experimental data and the $1/\sqrt{N}$ fit. For instance, at low photon numbers ($< 1750$), the 2D localizations are below the noise floor and hence the localization precision does not scale as $1/\sqrt{N}$. The discrepancies between the $1/\sqrt{N}$ and the 3D localization precision curves arise since this model does not account for factors which determine the localization precision in each perspective view throughout the depth of field - for instance pixel size, PSF extent and noise sources such as background and read noise for sCMOS cameras. The precision of the 3D fit depends on the optical model which combines the information from the 2D localizations each of which has different precision.
A

<figure><img src="image_19.png" /></figure>

B

<figure><img src="image_20.png" /></figure>

C

<figure><img src="image_21.png" /></figure>

D

<figure><img src="image_22.png" /></figure>

**Fig. S8.** Lateral and axial precision as a function of number of detected photons for each configuration. Top row contains data from configuration 1. Bottom row contains data from configuration 2. In each case a fitted line scaled as 1/√N has also been plotted.
<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /><figcaption>**Fig. S9.** Lateral and axial precision as a function of number of detected photons for each configuration. Top row contains data from configuration 1. Bottom row contains data from configuration 2. In each case a double exponential fit has also been plotted.</figcaption></figure>
## 2. SPHERICAL ABERRATION IN SMLFM

In the prototype system presented in this work, an oil immersion objective was used to image T-cells mounted in an aqueous medium ($n_s \approx 1.33$). The oil immersion lens was used as a result of the convenient size of its back focal plane, which was magnified using off-the-shelf achromatic lenses to a diameter approximately equal to an integer number of microlenses for both configurations. Oil immersion objectives are generally designed for stigmatic imaging of points located on the 'design surface', which typically lies immediately below the cover slip surface. Furthermore, stigmatic imaging only occurs when a certain thickness of immersion oil and cover slip are used. The results presented in Figure 6 in the main text were collected under experimental conditions depicted in Figure S10. Principally, emitters were located below a #1.5 coverslip, immersed in a medium with refractive index $n_s$, and imaged with an objective lens designed for an immersion medium with refractive index $n_{imm}$. In the ideal microscope described in Section 1, the phase in the pupil plane was considered entirely due to point source displacements from the focal position, described by Equation S4. According to this model, laterally displaced point sources on the nominal focal plane are imaged to identical locations in each perspective view in SMLFM. This is not the case when imaging into an aqueous medium using an oil immersion objective, a well characterized situation [3–6]. Following these treatments, consider a point source located at $(x_i, y_i, z_i) = (0, 0, 0)$, a distance $d$ below the cover slip. Figure S10 demonstrates how refraction of rays at the sample-cover slip interface results in a so-called 'focal-shift': the closest approximation to stigmatic imaging occurs at a plane located a distance $d$ below the coverslip when the $\rho^2$ terms of spherical aberration and defocus compensate one another.

Based on the assumptions that the surface of the front lens of the microscope objective is planar, and also that the microscope objective satisfies the sine condition, the phase in the back focal plane due to this focal-shift (or, spherical aberration) can be calculated by calculating the optical path difference $OPD_{sph.}$ between rays in the design system and in the experimental system which enter the microscope objective as parallel rays:

$$OPD_{sph.} = d(n_s \cos \varphi_2 - n_{imm.} \cos \varphi_1) \qquad (S16)$$

where, according to the sine condition:

$$\sin \varphi_1 = \frac{NA\rho}{n_{imm.}} \qquad (S17)$$

and, from Snell's law:

$$\sin \varphi_2 = \frac{NA\rho}{n_s} \qquad (S18)$$

such that the full expression for the phase in the pupil is:

$$\Phi_{sph.} = kd \left[ n_s \cos \left( \arcsin \left( \frac{NA\rho}{n_s} \right) \right) - n_{imm.} \cos \left( \arcsin \left( \frac{NA\rho}{n_{imm.}} \right) \right) \right] \qquad (S19)$$

with corresponding $\alpha_{sph.}$:

$$\alpha_{sph.} = 2d \left[ \frac{NA^2\rho^2}{n_{imm.} \sqrt{1 - \left(\frac{NA\rho}{n_{imm.}}\right)^2}} - \frac{NA^2\rho^2}{n_s \sqrt{1 - \left(\frac{NA\rho}{n_s}\right)^2}} \right] \qquad (S20)$$

As a result of spherical aberration, even a point emitter located at the nominal focal plane will be imaged to different locations in different perspective views in SMLFM. This is demonstrated in Figure S10, which depicts the phase in the back focal plane due to an emitter at the nominal focal plane. The total phase for emitters located at $(x_i, y_i, z_i)$ is now the sum of two terms:

$$\Phi_{tot.} = \Phi_{x_i, y_i, z_i} + \Phi_{sph.} \qquad (S21)$$

In other 3D SMLM approaches, it is necessary to estimate or calculate $\Phi_{sph.}$ in order to scale a calibration curve and correct the estimated $z_i$ positions. One such approach is phase retrieval, which requires a z stack of a bead on a cover slip to calculate the experimental phase [1, 2]. This is necessary because in most imaging modalities, angular information is lost when an intensity measurement is made using a detector. As described in the main text, in SMLFM, both intensity and angular information is captured. As a result it is possible to directly measure spherical aberration, and other aberrations, using the 2D localizations themselves. For all experimental data, these aberrations are estimated by measuring the average residual disparity across the field of view, for emitters located at the focal plane. The residual disparity is subsequently subtracted from all frames.

There are other subtleties to be considered when using an oil immersion lens to image into an aqueous medium. Firstly, the maximum solid angle, $\theta_{max}$ of the microscope objective lens ($\theta_{max} = 67^\circ$) is larger than the critical angle of refraction ($\theta_{crit.}$) between the aqueous medium and glass cover slip. As a result, for emitters located greater than a distance $\lambda_{em}$ away from the cover slip, the amplitude of the electric field above $\rho_{crit.}$ is zero in the pupil, where $\alpha(u, v)$ becomes complex [8–10]. This must be accounted for when calculating the average wavefront gradient across each microlens. In this proof-of-principle work, the average phase gradient was calculated numerically using $20^2$ points spanning each microlens as illustrated in (E) of Figure S1.

Another point to note is that refraction at the interface means that each point in the pupil plane receives light from a different solid angle than the case where the sample refractive index matches that of the immersion oil. Furthermore, polarization dependent Fresnel reflections at the sample-cover slip interface reduce the amplitude of the electric field in the pupil plane. Considering these losses is beyond the scope of this work, but would be important for the application of SMLFM to other single molecule experiments.
<figure><img src="image_27.png" /><figcaption>**Fig. S10.** (A) Imaging into aqueous medium ($n_s < n_{imm}$) using an oil immersion objective. Refraction at the boundary between the cover slip and the aqueous medium results in a focal shift. (B) For $\phi_{imm} > \theta_{crit}$, the amplitude of the electric field in the pupil plane is zero for emitters located $d > \lambda_{em}$ away from the cover slip. Configurations 1 and 2 were designed such that an integer number of microlenses (depicted by gray dashed lines) span this effective reduced pupil diameter.</figcaption></figure>

### 3. MLA ALIGNMENT FOR SMLFM

Mis-alignments of the microlens array can affect fit quality in SMLFM. Lateral mis-alignments are the least critical since they are equivalent to adding the same tilt to all wavefronts, corresponding to a global translation of all the images in the perspective views. It is however, important to ensure that the outer microlenses along the principal axes are not clipped, since this results in aberrated point spread functions which cannot be well approximated by 2D Gaussian functions. Furthermore, in cases where the paraxial approximation is insufficient, it is also necessary to estimate the location of the optical axis relative to the centre of the microlens array in order to accurately estimate emitter position. Lateral alignment of the MLA was carried out by using a 2D sample of fluorescent beads. Since, by design, in both configurations, an approximately integer number of microlenses span the magnified pupil, when the MLA is positioned correctly, the exterior microlenses are almost fully illuminated. Any lateral mis-alignments were clearly observable since they led to a radially asymmetric number of illuminated lenses as demonstrated in Figure S11.

Similarly to other point spread function engineering approaches, placing the microlens array in a plane other than a conjugate pupil plane will introduce errors since points from in different areas of the field of view will experience different phase modulation. The axial position of the MLA was determined by sending in collimated light through the microscope objective - corresponding to a focused beam in the BFP. The microlens array was placed in this plane as per [14].

<figure><img src="image_28.png" /><figcaption>**Fig. S11.** Lateral mis-alignments result in illumination of a radially asymmetric distribution of microlenses.</figcaption></figure>
Axial displacements of the sensor from the focal plane of the MLA decrease precision as photons are distributed over a larger number of pixels due to defocus. Furthermore, a given wavefront tilt will correspond to different $(x_{uv}, y_{uv})$ displacements than predicted by the optical model. To place the camera in the focal plane of the array, it was illuminated with a collimated beam. A focused array of spots was observed in the focal plane of the MLA, where the camera was placed.

Rotation of the microlens array is not intrinsically problematic, but the principal axes $(u, v)$ must be identified in order to accurately solve Equation S10. Rotating $(u, v)$ and maximizing the number of groups of 2D localizations which solved Equation ?? with low fit error as a function of the rotation angle $\theta_{\text{MLA}}$ was found to be an excellent angular discriminator.
## 4. SMLFM IMPLEMENTATION (HARDWARE)

All Experiments were performed on a customized, inverted microscope (Nikon, Eclipse Ti2). Illumination was provided by a set of lasers (405, 488, 561 and 638 nm) housed in an Omicron LightHUB. The output from each laser was coupled into a single mode optical fibre and collimated into an 9.6 mm diameter beam, using an off-axis parabolic mirror (Thorlabs RC08APC-P01). This beam was focused onto the back focal plane of a 60×, 1.42NA oil immersion objective (Olympus, PlanApo N) using a 300 mm focal length lens, $L_1$, to achieve wide-field excitation (96 µm FWHM). Where specified, data was acquired under Highly Inclined Laminated Optical (HILO) illumination [12], achieved by laterally displacing the excitation beam towards the edge of the back focal plane. A quad-band filter located between $L_1$ and the objective separated the incoming excitation beams from the collected fluorescence. A 200 mm tube lens, $L_2$ focused the collected fluorescence onto an intermediate image plane.

To perform SMLFM two more elements are required: another lens, $L_3$, placed in a 4f configuration with $L_2$ which relays the back focal (Fourier) plane onto the Microlens Array (MLA). The choice of the focal length of $L_3$ is crucial in determining the depth-of-field and efficiency of SMLFM for a given MLA. We used a square lattice MLA (SUSS micro-optics, 18-00178) with microlens pitch of 1015 µm and focal length of 25.4 mm. $L_3$ focal lengths of 75 mm and 100 mm were used to generate two different BFP diameter configurations (configuration 1 and configuration 2). These two configurations have differing number of illuminated microlenses and magnification factors to the detector (Hamamatsu, Flash 4.0 v2) which sits at the focal plane of the MLA.

<table><caption>**Table S2.** Comparison of characteristics of configuration 1 and configuration 2. NA<sub>eff</sub> refers to the effective numerical aperture of each microlens. Effective NA describes the portion of the NA of the microscope objective (which determines the extent of the pupil) bounded by each microlens. Resolution is stated according to the Abbe limit.</caption><thead><tr><th></th><th>Configuration 1</th><th>Configuration 2</th><th>Widefield (2D SMLM)</th></tr></thead><tbody><tr><td>Magnification</td><td>24.3</td><td>18.1</td><td>66.7</td></tr><tr><td>Pixel size (nm)</td><td>267.0</td><td>359</td><td>97.5</td></tr><tr><td>DOF (µm)</td><td>6.36</td><td>3.57</td><td>0.36</td></tr><tr><td>Max. efficiency (%)</td><td>66</td><td>87</td><td>100</td></tr><tr><td>NA<sub>eff</sub></td><td>0.45</td><td>0.34</td><td>-</td></tr><tr><td>Resolution (nm)</td><td>576</td><td>768.5</td><td>183</td></tr><tr><td>FOV (µm)</td><td>43.7</td><td>58.3</td><td>198.7</td></tr></tbody></table>
## 5. SUMMARY OF EXPERIMENTAL PARAMETERS

<table><caption>**Table S3.** Summary of parameters used to acquire data presented in Figure 4 of the main text. The header in each column refers to label of the figure.</caption><thead><tr><th>Parameter (units)</th><th>Fig. 4B</th><th>Fig. 4C</th><th>Fig. 4D (conf 1)</th><th>Fig. 4D (conf 2)</th><th>Fig. 4E</th><th>Fig. 4F</th></tr></thead><tbody><tr><td>λ (nm)</td><td>638</td><td>638</td><td>638</td><td>638</td><td>638</td><td>638</td></tr><tr><td>Avg no. photons</td><td>~ 3800</td><td>~ 4300</td><td>as plotted</td><td>as plotted</td><td>~ 3800</td><td>~ 4300</td></tr><tr><td>Exposure time (ms)</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td><td>10</td></tr><tr><td>No. frames per position</td><td>10</td><td>10</td><td>20</td><td>10</td><td>10</td><td>10</td></tr><tr><td>Z interval (nm)</td><td>10</td><td>10</td><td>100</td><td>50</td><td>10</td><td>10</td></tr><tr><td>Z range (μm)</td><td>4.7</td><td>6.5</td><td>4</td><td>7</td><td>4.7</td><td>6.5</td></tr></tbody></table>

<table><caption>**Table S4.** Summary of parameters used to acquire data presented in Figure 6 of the main text. The header in each column refers to label of the figure.</caption><thead><tr><th>Parameter (units)</th><th>Fig. 6A</th><th>Fig. 6C</th></tr></thead><tbody><tr><td>λ (nm)</td><td>638</td><td>638</td></tr><tr><td>Power (kW cm<sup>-2</sup>)</td><td>~ 0.8</td><td>~ 0.4</td></tr><tr><td>Exposure time (ms)</td><td>70</td><td>50</td></tr><tr><td>No. acquired frames</td><td>44676</td><td>150000</td></tr></tbody></table>
## 6. CRLB SIMULATIONS

To evaluate the theoretical performance of the two SMLFM configurations the Cramér-Rao lower bound (CRLB) was computed. The CRLB quantifies the fundamental limit of localization precision for an unbiased estimator according to:

$$ \sigma_{\theta} \geq \sqrt{\text{CRLB}_{\theta}} = \sqrt{\mathbf{I}^{-1}(\theta)} \qquad (S22) $$

where $\theta = (x, y, z)$ denotes the parameter to be estimated and $\mathbf{I}$ the Fisher information matrix. Following [16] and accounting for Poisson distributed shot noise, the diagonal elements of the Fisher information matrix are calculated as:

$$ F_{\theta} = \sum_{k=1}^{N_p} \frac{1}{\mu(k) + b} \left( \frac{\partial \mu(k)}{\partial \theta} \right)^2 \qquad (S23) $$

Here $N_p$ denotes the total number of sensor pixels, $\mu(k)$ the expected number of signal photons in the $k$th pixel and $b$ the expected background counts.

The point spread functions used to calculate the CRLB were simulated using parameters based on the experimental setup, namely, a 60×, 1.45NA oil-immersion objective lens ($f_{obj} = 3$ mm) and tube lens ($f_{tube} = 200$ mm) arranged in a 4f configuration, $\lambda_{em} = 630$ nm, 6.5 µm sensor pixel pitch, 6000 total detected photons and a constant background equivalent to 1 photon/384 nm (corresponding to the pixel size of configuration 2). The light field point spread function, $h_{LF}$, was simulated according to:

$$ h_{LF}(x_i, y_i, z_i) = \mathcal{F}^{-1} \left\{ \mathcal{F} \left[ \exp \left( i \left( \Phi_{MLA} + \Phi_{x_i, y_i, z_i} \right) \right) \right] \times \exp \left[ i 2 \pi f_{MLA} \sqrt{\left( \frac{1}{\lambda_{em}} \right)^2 - (f_s^2 + f_l^2)} \right] \right\} \qquad (S24) $$

where ($f_s, f_l$) represent the spatial frequencies in the camera plane, $\mathcal{F}$ and $\mathcal{F}^{-1}$, the Fourier Transform and Inverse Fourier Transform respectively, and $\Phi_{MLA}$, the phase imparted by the microlens array in the pupil plane, calculated according to:

$$ \Phi_{MLA}(\rho) = \left[ \text{rect}\left(\frac{\rho}{d}\right) \exp\left(\frac{-ik}{2f_{MLA}}\right) \right] \otimes \text{comb}\left(\frac{\rho}{d}\right) \qquad (S25) $$

For the point spread function engineering approaches, the point spread function was calculated as the Fourier transform of the electric field in the objective pupil plane:

$$ \mathbf{E}_{BFP} = \exp \left\{ i \left( n_s z_i k \sqrt{1 - \left( \frac{\text{NA}\rho}{n_s} \right)^2} + \Phi_{\text{PSF}} \right) \right\} \qquad (S26) $$

where $\Phi_{PSF}$ is the phase mask specific to the engineered PSF, for instance in the case of the Tetrapod or saddle point:

$$ \Phi_{\text{Tetrapod}} = 0.5\sqrt{6\rho^2} \cos 2\theta - 0.65\sqrt{10}(4\rho^4 - 3\rho^2) \cos 2\theta \qquad (S27) $$

which was also re-scaled to range between $[-\pi, \pi]$ following [17]. The phase mask commonly used to generate double helix point spread functions was read from an image file, based on the method presented in [19].
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