Three-dimensional nanometre localization of nanoparticles to enhance super-resolution microscopy

Meeting the nanometre resolution promised by super-resolution microscopy techniques (pointillist: PALM, STORM, scanning: STED) requires stabilizing the sample drifts in real time during the whole acquisition process. Metal nanoparticles are excellent probes to track the lateral drifts as they provide crisp and photostable information. However, achieving nanometre axial super-localization is still a major challenge, as diffraction imposes large depths-of-fields. Here we demonstrate fast full three-dimensional nanometre super-localization of gold nanoparticles through simultaneous intensity and phase imaging with a wavefront-sensing camera based on quadriwave lateral shearing interferometry. We show how to combine the intensity and phase information to provide the key to the third axial dimension. Presently, we demonstrate even in the occurrence of large three-dimensional fluctuations of several microns, unprecedented sub-nanometre localization accuracies down to 0.7 nm in lateral and 2.7 nm in axial directions at 50 frames per second. We demonstrate that nanoscale stabilization greatly enhances the image quality and resolution in direct stochastic optical reconstruction microscopy imaging.

(d) Image resolution gain with 3D drift correction ( Figure 4: Comparison of the dynamic axial accuracy of our approach with a commercial autofocus system (Nikon Perfect Focus). The autofocus system controls the microscope displacement (black dots), while our method quantifies the actual axial position of a 100 nm gold nanoparticle (red line). Emerald stripes indicate the intervals where the autofocus system is switched on (note that here the stage is heated to speed the axial drift).    Fig. 2.b,pink) Lateral (xy)+low threshold S. Fig. 1.c 128222 > 33 (S. Fig. 2.b,yellow) Axial (z) Data of S.

Supplementary Discussion
Theoretical description of the through-focus dependencies The intensity and the phase are not independent variables respect to the imaging plane. They are indeed in quadrature, as one can observe for example on the Fig. 1c: when the slope of the intensity versus z is null, the phase versus focus is important and vice versa. The so-called transport-of-intensity equations can be used to describe the link between the intensity and phase as a function of the axial plane [1]. Let u be the amplitude of the EM field: with r the lateral (x, y) coordinates and z the axial coordinate. I represents the EM field intensity and ϕ its phase. It is noteworthy that with our phase-imaging approach (QWLSI) this is actually the optical path difference (OPD, denoted δ in the further equations) that is measured. The OPD is linked to phase with: with k = 2π/λ the wavevector norm and ϕ prop the propagation phase defined as ϕ prop = k · z.
Considering a free-space propagation of a coherent beam, one can write [1] j ∂ ∂z with ∇ the lateral gradient operator. Combining the equ. (1) and (3) we obtain Useful relations between I and ϕ can be reached by multiplying the equ. (4) by the complex conjugate of u * . A second equation can be generated by taking the complex conjugate of (4)·u * . By adding or subtracting these two equations, simple links between the axial phase or intensity derivative and their lateral gradients can be obtained: The first part of Eq.(5) is well known in the field of quantitative phase imaging as numerous techniques are actually based on this equation to retrieve the phase using intensity measurements [2,3,4]. The second part is less known as it is usually simpler to measure the intensity rather than the phase. Let us use both of them in the scope of non resolved absorbing sample imaging (ex. gold NPs). In this case the in-focus intensity and phase images can be modeled as: considering for simplification purpose that the imaging response of the optical microscope can be described by a 2D Gaussian function with standard deviation σ. A is the maximum intensity attenuation value at the particle image center and I 0 the background intensity. As the object is purely absorbing, its phase at the plane z = 0 is constant equal to ϕ prop = 0. Using the equ. (5) and (6) one can obtain: By considering the microscope as aberration-free with an objective numerical aperture of NA obj = 1.4, it means that σ = r P SF 2 √ 2 ln 2 = 0.26λ NA obj . At the NP image center (i.e. r = 0) the equ. (7) can be rewritten in OPD slope: As experimentally observed, the intensity derivative along z is null when the particle is at the focus ( Fig. 1c in black). Simultaneously, the OPD derivative along z is constant confirming the linear z dependency of the measured phase around the focus (Fig. 1c in red). An interesting point is that the OPD slope is independent of λ, the key parameter being the attenuation A: it means that our technique can be used at any wavelength only choosing a NP that exhibits a strong attenuation at this particular wavelength.
Moreover, the phase slope values (near the focus) retrieved with this model are in good agreement with the measured ones (Suppl. Tab. 1).

Image resolution gain with 3D drift correction
The lateral super-resolution for pointillist techniques depends on multiple aspects: (i) the lateral drifts during the acquisition, (ii) the number of detected molecules (Nyquist criterion), and (iii) the localization precision which is mainly affected by the fluorophore brightness. We infer the spatial resolution in dSTORM imaging by measuring the histogram of the detected positions along a single nanostructure. Tubulin and actin filaments are good candidates, and are commonly used to determine the resolution in super-resolution experiments [5,6,7]. One limitation of this method is that the measured histogram width is a convolution of the object by the microscope point spread function. Hence, the histogram width depend on the size of the object. Typically, the smallest structures with isolated F-actin filaments labeled using phalloidin have a diameter below 10 nm, while tubulin filaments labeled with primary and secondary anti-bodies have a diameter around 50 nm. Therefore, the histogram widths and standard deviations measured on F-actin filaments are narrower than those obtained on tubulin.
Suppl. Figure 2 and Suppl. Table 2 show the results obtained using the data from the experiments described in Fig. 3 and Suppl. Fig. 1. The lateral resolution is significantly better with (x, y, z) drift compensation up to a factor of 3 in our F-actin experiment. The lateral resolution, in terms of standard deviation of the position histogram, is measured to be 11±1 nm with F-actin labeled with phalloidin plus 3D drift compensation, while the resolution for the tubulin labeling experiment and 3D drift correction is 21±1 nm. In addition to the lower spatial resolution when no drift compensation is active, the number of detected molecules is also 3 times lower as compared to a full-3D drift compensation. This comes as the defocus accumulated during the experiment blurs the fluorescence intensity below the detection threshold. Decreasing the detection threshold in the absence of axial stabilization helps to increase the number of detected molecules. However, as the detection signalto-noise ratio is low for many fluorophores, it leads to a high localization uncertainty and thus a bad lateral resolution although (x, y) drift correction is applied.
We concentrate here on the axial localization dynamics and compare our method with a commercial autofocus system (Perfect Focus, Nikon, Japan) which monitors the reflection of infrared light at the coverslip surface. The single output from this commercial solution is the actual microscope axial position with an accuracy of 25 nm. Therefore, to test our technique, we decided to sequentially activate and deactivate the autofocus system while measuring with our approach the relative axial displacement of a 100 nm gold nanoparticle deposited on a coverslip and immersed in water (observation wavelength 610 nm). In order to increase the drift speed, the sample was heated using a commercial sample holder (Tokai Hit, Tokai, Japan).
Suppl. Figure 4 shows the result of this comparison. When the commercial autofocus is on (emerald stripes), our technique measures a residual nanoparticle axial drift that is not detected with the autofocus, or improperly corrected (see the dispersion of black dots when the autofocus is on as compared to the actual nanoparticle position). When the autofocus is turned off (white stripes), the focus drift is no longer compensated by the microscope mechanical displacement. When the autofocus is turned on again, the microscope makes a displacement whose amplitude is correct with the measured nanoparticle axial drift. In all cases, it appears that our localization approach provides a better spatial and temporal accuracy. Moreover, we point out that autofocus systems based on the reflection of light at the coverslip/medium interface are inefficient in the case of index matching between the coverslip and the medium (as for example for in-depth STED measurements), which is not the case for our technique.
In the theoretical description of the through-focus intensity and phase imaging (Section 3), we have considered the microscope objective as being perfectly corrected from optical aberrations. This assumption is used to simplify the theoretical treatment, but it is not a necessary condition for our technique. Especially at high numerical aperture such as 1.4 or even 1.49 used for super-resolution imaging, the microscope objective might have optical aberrations. One of the major advantage of our technique is that there is no assumption on the optical response of the imaged nanoparticles. As both the intensity and the phase are simultaneously measured, the optical aberrations are directly taken into account in the numerical propagation algorithm. High optical aberrations will only result in a drop in our positioning sensitivity, but measurements are still possible. Figure 2.b,c and Movie 1 illustrate the accuracy of refocusing with large defocus and thus spherical aberrations. For a defocus of several microns, the refocused intensity PSF is enlarged which leads to a reduction of lateral localization precision, and the trough-focus phase slope is decreased leading to a drop in axial localization sensitivity. Remarkably, this drop in localization accuracy remains limited, even for defocus much larger than the depth-of-field. For a 3 µm defocus, we typically achieve lateral precision below 10 nm and axial precision below 40 nm (Fig. 2c).
Within the wavefront sensor field of view (FoV), one way to experimentally evaluate the influence of field aberrations is to compare the 3D localization of three different immobilized particles: 2 near the center of the FoV (particle 1 and 2) and 1 near the edge (particle 3), as presented in Suppl.  Table 3 summarizes the standard deviation for 100 measurements of the difference of positioning for each couple of particles, along both the lateral x and the axial z direction. There is a small effect of the field aberrations that tends to slightly decrease the measurement precision on the edge of the FoV. However, this variation is very limited (less than 1 nm) and can be safely neglected. This shows that any nanoparticle(s) within the FoV can be considered to stabilize the setup.
To optimize the accuracy of our 3D position retrieval technique, it is useful to maximize the interaction of light with the nanoparticle. This can be elegantly performed by working close to the local surface plasmon resonance (LSPR) of the nanoparticle [8]. Near the LSPR, the extinction cross-section is remarkably high due to the light that is resonantly absorbed and scattered by the nanoparticle. One consequence is that the through-focus intensity and phase responses are respectively even and odd with important slopes and extremum values near the LSPR wavelength. Suppl. Figure 6a shows throughfocus intensity and phase values for 100 nm gold NP at two different wavelengths λ ≈ 445 nm and λ ≈ 607 nm, respectively off and near resonance conditions. Using numerical computations according to the Mie theory, the extinction cross section of a 100 nm gold sphere exhibits a resonance around λ = 580 nm (Suppl. Fig. 6b). This is visible on the through-focus intensity and phase responses: the curves are more modulated at 607 nm (solid orange line in Suppl. Fig. 6a) compared to 445 nm (dashed blue line in Suppl. Fig. 6). Our 3D superlocalization approach is thus more sensitive for this kind of particle around 580 nm.
In this study, we use spherical nanoparticles and non-polarized illumination from a collimated halogen lamp. As the nanoparticles are isotropic, the illumination polarization has no influence on our results. However, if non 3D isotropic particles such as metal nanorods were used for our 3D localization approach, the illumination polarization would need to be tuned to maximize the sensitivity of our technique. This is due to the enhancement of interaction between the nanoparticle and the illumination light.

Comparison between metal and dielectric nanoparticles
So far, we considered 3D super-localization of gold nanoparticles which have a high extinction, and whose optical response is dominated by the imaginary part of the gold refractive index. To generalize our approach, we investigate here dielectric nanoparticles (such as latex or silica NPs) where the optical response primarily depends on the real part of the particle refractive index. Suppl. Figure 7 shows a comparison between gold and latex nanoparticles. At the microscope focus, the latex nanoparticles are almost invisible in intensity while they largely stand out of the background in the phase image. Therefore the phase image can be used to laterally localize the latex nanoparticles, while the intensity variation can be used to obtain the axial position. Interestingly, the shapes of the phase and intensity axial responses are inverted between metal and dielectric nanoparticles (Suppl. Fig. 7c,d). While gold nanoparticles provide a sharper response due to their higher extinction, our technique can nevertheless be applied to dielectric nanoparticles. Notice also that the two nanoparticles are localized by our technique in different planes distant of 150 nm. This is consistent with the fact that the latex particles are larger than the gold ones by 150 nm.

Numerical propagation and refocusing
As the illumination can be considered as a non-polarized plane-wave, it is possible to numerically propagate an acquired scalar electromagnetic field in different planes. It generates images in various planes without any moving elements. Indeed, under coherent illumination, the sample imaged frequencies K o are carried by a portion of the Ewald sphere [9,10]. If the illumination wavevector − → k i is collinear to the optical axis, then it is straightforward to obtain a representation of K o in a trans-illumination scheme, given the fact that diffraction is an elastic process: with − → k d a diffracted wavevector.
A phase and intensity measurement in a single 2D plane (ex. Suppl. Fig. 8a) is enough to regenerate the 3D frequency space (Suppl. Fig. 8c) by projecting the frequency content of the measured 2D electromagnetic field onto the 3D Ewald sphere (Suppl. Fig. 8b). This allows to retrieve the phase and the intensity images in different planes by a simple inverse Fourier Transform (FT).

Accuracy measurement protocol
Each measurement p of the nanoparticle position is the sum of the sample displacement d plus the error of our approach ε. In order to measure the accuracy of our approach, that is the standard deviation of the error σ(ε), we performed 100 repeated measurements of the nanoparticle position at 50 Hz rate. The total 2s integration time implies that during the recording some technical noise will also affect the sample displacement d, which is not a constant. To assess only the accuracy of our superlocalization method σ(ε), we subtract the measured position of two immobilized nanoparticles visible within the same image (Fig. 1b). This removes the correlated noise σ(d) of the actual sample displacement leaving only the combined noise of each localization measurement: with p i , ε i and d respectively the i th particle measured position, the error and the real drift. As the position measurement of each particle is driven by independent but equivalent noise, it means that the mean square fluctuation of each particle σ (p i ) = σ (ǫ i ) is the same for each particle (i.e. σ (ǫ 1 ) = σ (ǫ 2 )) and thus: With the technique applied on 100 nm gold nanobeads at λ = 594 nm, the accuracy from the comparative positions of two nanoparticles has been found to σ (ε xy ) = 0.7 nm in the lateral directions and σ (ε z ) = 2.7 nm in the axial direction when the particle is near the focus.
It is interesting to note that this approach allows to quantify the fluctuations of the setup stability σ(d) on a 2 seconds timescale: Using the data in Fig. 2b for σ(p) and the noise estimates for σ(ε) from the correlated positions of 2 nanoparticles, we obtain the setup lateral stability σ(d xy ) = 1.3 nm while the axial stability is σ(d z ) = 6 nm. These fluctuations originate mainly from the noise of the piezo stage and vibrations from the wavefront sensing camera. Both are independent of our conceptual approach.

Super-resolved image acquisition and reconstruction
We acquired 2000 to 3500 blinking images for each super-resolved reconstructed image. The axial defocus is compensated in real time, by moving the objective, using the z displacement measured with our approach. On each image, a local threshold τ is applied to roughly detect each point spread function (PSF). Each PSF is then fitted with a Gaussian curve using the Levenberg-Marquardt nonlinear algorithm on a home-made Labview software (LV2011, National Instrument, USA) to measure the particle centroid. The lateral drift is compensated after the acquisition by subtracting the measured lateral displacement of the reference NP to each measured centroid. In order to take into account the signal-to-noise ratio of each detected fluorophore, the super-resolved image is created by adding a 2D Gaussian function centered on each detected particle centroid, with a standard deviation depending on the fluorescence signal-to-noise ratio (SNR) for this particle.
To generate the super-resolved images when no axial stabilization is applied (ex. Fig. 3b,c or Suppl. Fig. 1b,c), the same set of camera images and thus detected particles are used. Prior to the dSTORM experiments, the microscope PSF has been calibrated by acquiring images of individual fluorescent nanobeads (50 nm diameter) and measuring the signal variation as a function of the axial defocus. The effects of the sample axial drift without axial stabilization are obtained as follows: (i) each fluorophore image position, energy and background is measured via 2D Gaussian fitting on the acquired stabilized image. (ii) A new fluorophore subimage is then numerically generated for each detected molecule, respecting the energy conservation, using the experimental microscope PSF corresponding to the total axial defocus that would have been present in the absence of stabilization. (iii) The background is added to the subimage and Poisson noise is computed on each pixel given this new value and the gray-level to photon conversion factor of the camera. (iv) If some pixels in the generated final subimage is above the threshold τ , the particle is still considered as detected and its 2D localization is computed. (v) The super-resolved image without axial defocus is generated as previously described, by the addition of each detected molecule.