Three-dimensional single-molecule localization with nanometer accuracy using Metal-Induced Energy Transfer (MIET) imaging

Single-molecule localization based super-resolution microscopy has become an important tool in biological imaging. We would like to mention here two of the most successful examples of its application: using high-resolution 3D-Stochastic Optical Reconstruction Microscopy (STORM), Xu et al.¹ found periodic cytoskeleton structures in neuronal axons, and by pushing the limits of direct STORM (dSTORM), Löschberger et al.² could resolve the eightfold radially symmetric arrangement of an integral membrane protein, gp210, in a nuclear pore complex. Although STORM/dSTORM primarily improves the resolution in the lateral direction (xy-plane), techniques such as astigmatism-based imaging,³ biplane imaging,⁴ or helical wavefront shaping⁵ enabled the generation of images with full three-dimensional resolution. However, for all these techniques, the achievable axial resolution (50 nm) is still approximately one order of magnitude less than typical intramolecular distances. Thus, until now, they were not much used for structural biology on the single macromolecule level. There is, however, one class of techniques which indeed achieves single nanometer localization accuracy along the optical axis, and it is based on light interferometry. The most prominent members of this class are interferometric Photoactivated Localization Microscopy (iPALM)⁶ and 4pi-STORM.⁷ As only one remarkable example of their capability, we cite here the work by Kanchanawong et al.⁸ who used iPALM in conjunction with specialized photo-activatable fluorescent proteins fused to focal adhesion proteins in U2OS cells to localize different proteins in the focal adhesion cluster with a resolution of 10 nm–15 nm along the optical axis. However, the main challenge of these interferometry-based methods is their enormous technical complexity, which has so far prevented their broad applicability.

Recently, we introduced a novel single-molecule fluorescence-spectroscopic method, single-molecule Metal-Induced Energy Transfer (smMIET), that allows for determining the distance of an emitter from a metal surface with nanometer accuracy.⁹ It is based on the distance-dependent electromagnetic coupling of a fluorescent emitter to surface plasmons in a metal, which modulates its excited-state lifetime. By measuring the fluorescence lifetime of an emitter, one can determine its axial position with nanometer accuracy. Thus, smMIET is based on the conversion of temporal information into spatial information. In this work, we outline the theoretical framework of smMIET and discuss its potential, in combination with single-molecule localization based super-resolution methods, for achieving isotropic nanometer localization accuracy. We present first experimental results for molecules deposited directly on the surface of a flat silica spacer or embedded into a thin polymer film. Our results demonstrate that one can indeed achieve nearly isotropic single-nanometer single-molecule localization accuracy, by using an experimental setup of reasonable technical complexity.

Almost all fluorescent emitters are well described by the model of an ideal oscillating electric dipole in a semi-classical framework. The near-field coupling of an emitting electric dipole and a planar multilayered substrate has been thoroughly described in several publications, see for example, Refs. 10 and 11. In short, the emitted electromagnetic field of the electric dipole is expressed mathematically as a superposition of plane waves. For each plane wave, the interaction with the layered substrate is calculated by using standard Fresnel theory. In the end, this gives us the full electromagnetic field of the dipole in the presence of the multilayered substrate. Next, by calculating the electromagnetic energy flux (Poynting vector) into the solid angle sin θdθdϕ, we obtain the angular distribution of radiation (the probability that a photon is emitted into a given solid angle element) of the dipole. Finally, by integrating the Poynting vector over a given surface, we calculate the total energy flux of the field emitted by the dipole through the surface. This approach can be used for finding the full radiation power of the dipole emitter as well as the part of the energy that is absorbed by the multilayered substrate (if it is not purely dielectric). Furthermore, the far-field emission of the dipole into the cone of light collection of an objective can be determined which allows us to estimate the efficiency with which a molecule is detected using a fluorescence microscope (collection efficiency function). The details of all these calculations are presented in the supplementary material.

Throughout our current paper, we consider the following particular sample geometry: The substrate above which fluorescent molecules are immobilized consists of a thin semi-transparent metal film deposited on a glass support and topped with a thin transparent silica spacer. In our work, we mostly use a 10 nm gold film sandwiched between 2 nm and 1 nm titanium layers above and below, evaporated on top of a standard glass coverslip (thickness = 170 μm, refractive index = 1.52). We refer to this layered metallic substrate as the MIET substrate in this article. A silica spacer of chosen thickness is evaporated on top of the metal layers to avoid direct contact of a fluorophore, which would lead to complete quenching of fluorescence. A fluorescent molecule, described by its dipole-moment amplitude vector by p, is positioned in a medium with refractive index n₁ at a position r₀ = (ρ₀, z₀) above the silica interface in an infinite homogeneous space with refractive index n₁. See Fig. 1 for the geometry of the dipole’s emission above an interface.

We write here the end result for the total field at the dipole’s position that is given by

According to Poynting’s theorem, the work per time done on the dipole by the electric field (and thus the dipole’s emission rate) is given by

where ω = ck₀ is the oscillation frequency of the dipole. Let us now consider the special case of a vertical dipole, $p=pe^z$⁠. In that case, we find

where we have taken into account that $(e^1p+⋅e^z)=(e^1p−⋅e^z)=q/k1$⁠. Similarly, for a parallel dipole, $p=pe^x$⁠, we find

where we have taken into account that $(e^1p+⋅e^x)=−(e^1p−⋅e^x)=w1/k1⁡cos⁡ψ$ and $(e^s⋅e^x)=sinψ$⁠, with ψ being the angle between the x-axis and the vector q, and furthermore, that the average of cos²ψ and sin²ψ over the full circle is both 1/2. Finally, the emission rate of a dipole oriented at an arbitrary angle α with respect to the normal of the surface and at height z₀ can be written as

To check the validity of all expressions, let us consider the case that there is no interface, i.e., let us inspect the derived expressions for the special case of R_p ≡ R_s ≡ 0. Due to taking the real part in Eqs. (3) and (4), we have only to consider the integral up to the upper bound q = k₁ because the integrand becomes purely imaginary for larger values of q. Then, setting q = k₁ sin θ and $w1$ = k₁ cos θ, we find from Eq. (3)

where S₀ is the emission rate of the free dipole in a homogeneous medium with refractive index n₁. The same result is obtained when starting from Eq. (4).

All our expressions so far are not only true for a single interface, but also for any stratified conducting/dielectric layer system, as long as one uses effective reflection and transmission coefficients.¹² In particular, for a thin metal film (with complex-valued refractive index n_m and thickness h) sandwiched between glass (refractive index n₂) and air (refractive index n₁), one has

and

where the subscripts refer to p- and s-polarization, respectively, and the $rp,sjk$ and $tp,sjk$ are Fresnel’s reflection and transmission coefficients for a single interface dividing medium j from medium k, and $wm=nm2k02−q2$⁠. See for example Fig. S2 of the supplementary material for calculated reflection and transmission coefficients.

Knowing the total emission rates of the dipole allows now for calculating the observable fluorescence lifetime of a real fluorescent (electric dipole) emitter. Ideally, this fluorescence lifetime will be inversely proportional to the just calculated emission rates. However, real fluorescent dyes have a fluorescent quantum yield ϕ which is typically lower than one, the ratio between the fluorescence lifetime in the presence of an interface and the lifetime within a homogeneous medium without the interface is given by

where we have taken into account that the radiative emission rate S in front of an interface is orientation- and position-dependent. S₀ is the radiative emission rate in free space with the same refractive index and far away from any dielectric or metallic interfaces. Often, we are interested in the lifetime of a dye which can freely rotate, with a rotational diffusion time much faster than the typical fluorescence decay time. In that case, one has to average the decay rate, Eq. (5), over all orientations, which then defines an orientation-averaged and only position-dependent fluorescence decay time τ_f(z) via

As an example, we calculated the fluorescence lifetime of an emitter at 690 nm emission wavelength as a function of distance and quantum yield. The emitter is placed in water (refractive index n₁ = 1.33); the sample consists of glass (refractive index n₂ = 1.52) which is covered by 2 nm titanium, 10 nm thick thin gold film, 1 nm titanium (complex-valued refractive index n_Au = 0.17 + 3.79i, n_Ti = 2.18 + 3.27i, at λ_em = 690 nm) that is topped with a 10 nm thick layer of SiO₂ (refractive index $nSiO2=1.46$⁠). The result is shown in Fig. 2 for ten different values of quantum yield ϕ. This position-dependent fluorescence lifetime is the fundamental basis of MIET: it establishes a direct relationship between measurable fluorescence lifetime and vertical position of the fluorescing molecule above a surface.

Besides the position-dependent fluorescence lifetime, another important parameter of a MIET experiment is the efficiency with which a molecule can be detected when using an epi-fluorescence microscope which looks on the molecule from below (through the metal film). For example, the total emission power of a dipole emitter into the cone of light detection of an objective is given by (for details, see Subsection B in Sec. S1 of the supplementary material)

where Θ is the maximum half-angle of the objective’s light collection cone. In the case of a rapidly rotating molecule (rotational diffusion time much shorter than the fluorescence decay time), the average collection efficiency N_↓,↑(z₀) is found by averaging N_↓,↑(α, z₀) over α, $N↓,↑(z0)=⟨N↓,↑(α,z0)⟩α$⁠.

As an example, we performed calculations for a dipole with same emission wavelength, sample geometry, and parameters as used in Fig. 2. Figure 3 shows the distribution of relative emission into each half-space [by setting Θ = π/2 in Eq. (11) and the corresponding equation for N_↑], separately for a horizontal and a vertical dipole orientation and as a function of distance z₀ from the SiO₂ surface. The fluorophore is almost quenched when placed directly on top of the 10 nm spacer, even for such a small metal film thickness of only 10 nm. Furthermore, as can be seen from the green curve in the inset, around 2/3 of a (randomly oriented) dipole’s total emission passes indeed through the glass gold layer and can be collected from the bottom. This result clearly favors detection from the glass side in contrast to detection from above (see also Ref. 11). The experimental proof that it is indeed possible to efficiently detect single molecules through a metal film (despite unavoidable absorption losses in the metal) was first demonstrated by Stefani et al.¹³ 

To obtain better insight into how much of the light emitted into the bottom half-space can be indeed collected by an objective, we calculated the angular distribution of emission (averaged over all possible dipole orientations) also as a function of the distance of the dipole from the surface. The result is shown in Fig. 4. The bottom-left subplot shows also the different cones of light collection for different N.A. values. All other sub-plots of angular distribution of emission show only the cone of light collection for the highest considered N.A. of 1.49. As can be seen, most of the radiation is emitted into the glass substrate at high emission angles, which requires a high N.A. objective to collect this emission efficiently. The bottom-right subplot shows the collection efficiency N_↓(z₀) for different dipole distances z₀ in relation to several values of N.A. Finally, Fig. 5 shows the proportion of the light collected for a rapidly rotating dipole, i.e., relative brightness as a function of distance from the substrate. The plot also shows the collection efficiency achieved by an objective with high numerical aperture (N.A. = 1.49). As can be seen, more than 95% of the radiation is collected by the objective irrespective of the distance of the dipole from the substrate.

The above model calculations were mostly concerned with the case of a randomly oriented or rapidly rotating emitter, which is the case most frequently encountered in experiments. However, as can be seen from Eq. (5), the emission rate of an emitter above an interface is, in general, orientation-dependent. Thus, for a non-rotating emitter with fixed dipole orientation, it is necessary to know its three-dimensional orientation (in particular its out-of-plane angle) for correctly translating a measured fluorescence lifetime value into a distance value. As an example, let us consider an emitter embedded within a thin polymer film of 20 nm thickness above the substrate, emitting at λ_em = 542 nm with quantum yield ϕ = 0.8 and free-space fluorescence lifetime (τ₀) of 3 ns. Figure 6 shows the calculated fluorescence lifetime values as functions of the axial position z₀ for various orientations. As can be seen, assuming random orientations may lead to errors as large as ∼10 nm in the estimation of the axial positions of the molecules.

Several methods exist for determining the three-dimensional orientation of dipole emitters above an interface. Here, we will discuss two of them which are particularly suited for being combined with smMIET: taking scan images with a radially polarized laser beam, and defocused imaging with a wide-field setup. A theoretical basis of both methods is presented in Secs. S2 and S3 of the supplementary material. Briefly, scanning an immobilized molecule with a radially polarized beam yields characteristic intensity patterns that reflect the orientation of its excitation dipole. This is due to the peculiar three-dimensional electric field polarization structure that is produced in the focus.¹⁴ By contrast, defocused imaging (imaging with a purposefully defocused microscope) uses the effect of the dramatically altered and orientation-dependent angular distribution of radiation from a dipole close to an interface. This shows up as blurred but well-defined structured intensity patterns in the defocused image. Exact wave-optical calculations allow us to precisely model the intensity patterns as a function of dipole orientation for both these methods (see for example Ref. 15 for the case of defocused imaging). A least-squares minimization pattern-matching method can be used to estimate the orientations of the dipoles using a discrete set of pre-calculated intensity patterns for different emitter orientations¹⁶ (see also Sec. S4 of the supplementary material). These orientations can then be used as initial guess values for further orientation refinement by minimizing a log-likelihood function Λ that takes into account the Poissonian statistics of photon detection,^17,18

Here, P(r|α, β, r₀) denotes the calculated excitation/emission intensity patterns (see Secs. S2 and S3 of the supplementary material). These patterns depend on the molecule’s orientation angles α and β, and its lateral position r₀. The parameters A and B account for the molecule’s brightness (the number of detected photons) and a (uniform) background signal. Thus, one minimizes the log likelihood function with six free parameters (orientation, lateral position, brightness, and background), which can be carried out by using, e.g., a conjugate gradient method.

In what follows, we will estimate lower bounds of the three-dimensional localization uncertainty of single emitters with fixed orientation when using these methods for orientation determination. Furthermore, we will also analyze the applicability of these two methods in the case when the dipoles are rapidly rotating.

In this section, we present a theoretical analysis of smMIET imaging for 3D-localization of a single molecule, i.e., how accurately one can determine the position of a single molecule in all three dimensions. This obviously depends on the number of photons that one can detect until photo-bleaching (or photo-switching). The first step toward estimating the position of a molecule is to localize it in the lateral plane. One way to do this is by recording scan images using a linearly polarized beam, and then performing a weighted least-squares fitting of the intensity distribution with a two-dimensional (2D) Gaussian. The lower bound of the localization precision is given by the standard deviation of the point spread function σ_PSF divided by $N$⁠, where N is the number of recorded photons. In the presence of background B and finite pixel size a ≤ σ_PSF, this lower bound is found as¹⁷ 

For a high signal-to-noise ratio, this reduces to

Next, we estimate the axial localization accuracy using smMIET. In smMIET, the height of a single molecule above a substrate is determined by converting its lifetime value onto a height value using the known lifetime vs. height curve, as shown, e.g., in Fig. 2. The accuracy of this height estimation is governed by the error of the lifetime estimation Δτ and is given by

Assuming a Poisson photon-detection statistics, the uncertainty of estimating the fluorescence lifetime Δτ from a recorded fluorescence decay curve with N photons is ideally given by

Therefore, in order to minimize the axial localization error, one has to collect as many photons from an emitter as possible. An efficient way to achieve this is by localizing a molecule based on a theoretical point spread function (PSF) model (2D Gaussian in this case), identifying all the pixels corresponding to that molecule, collecting all the photons from the identified pixels, and finally estimating their fluorescence lifetime by fitting a mono-exponential decay. This significantly reduces the uncertainty in estimating the fluorescence lifetime as compared to calculating the lifetime for each individual pixel.

Using Eqs. (14) and (15), one can now estimate the lower bounds of simultaneous axial and lateral localization accuracy of an emitter if, on average, N photons are collected. When looking at the number of detectable photons, one has not only to consider the collection efficiency as shown in Fig. 5, but also the fact that the presence of the metal film enhances the photostability of a dye because the average time spent by a molecule in its excited state is reduced by the presence of the metal surface.¹⁹ As an example, let us assume that one collects approximately 10⁴ photons from a single molecule before photobleaching if it has a quantum yield of one. Then, Fig. 7 shows a comparison of the average number of detectable photons as a function of distance from the surface and for different quantum yield values taking into account both the distance-dependent collection efficiency and photostability.

Using Eqs. (16) and (15), this can be translated into localization accuracy curves as shown in Fig. 8. These calculations show that a dye molecule with a quantum yield value of ϕ ≥ 0.3 can be localized with an uncertainty of ∼5 nm within a distance range between 10 nm and 60 nm from the substrate. This requires the collection of only as few as 700–1000 photons from the molecule. If one considers an emitter with quantum yield 0.5, the localization accuracy becomes better than 3 nm in a distance range from 20 nm to 60 nm.

In Subsection II D 1, we considered the localization accuracy assuming a rapidly rotating emitter that is scanned with a linearly polarized laser for determining the lateral position, and the dipole-orientation averaged distance-lifetime curves for estimating its axial position. The situation becomes more complicated for a fixed dipole orientation, in which case one has also to determine the orientation of the molecule for being able to calculate its correct axial position from a measured lifetime value. As discussed in Sec. II C, there exist basically two methods for measuring the orientation: scanning with a radially polarized laser focus, or imaging with a defocused wide-field microscope. In this subsection, we consider the first method. The advantage of scanning with a radially focused laser beam is that one can simultaneously, in one measurement, determine the three-dimensional orientation and the fluorescence lifetime. As before, one first localizes a molecule laterally, then bundles all detected photons to estimate the fluorescence lifetime, and finally uses the orientation and lifetime information to determine the axial position.

The uncertainties of both the fluorescence lifetime and the inclination angle then contribute to the final axial localization error as

The first part of the error is due to the uncertainty in estimating the fluorescence lifetime, and the second part is due to the uncertainty of determining the dipole orientation. As an example, we consider again the emitter from Sec. II C with λ_em = 542 nm, ϕ = 0.8, embedded in a thin polymer film (refractive index ≈1.5, thickness of 20 nm) on top of the metallic substrate. We estimate the axial localization error due to the uncertainty of determining the lifetimes in the following way: First, we calculate the fluorescence lifetime of the emitter as a function of its axial position z₀ and orientation α using Eqs. (5) and (9). Next, we determine the molecular detection function of the dipole, that is the product of the excitation probability, taking the peculiar electric field polarization structure in the focus of the radially polarized laser, together with the collection efficiency function given by Eq. (11). Now all that remains is to calculate the lower bound of the fluorescence lifetime uncertainty Δτ using Eq. (16). For fixing the number of detected photons, we assume that one collects on average 10³ photons when measuring a rapidly rotating dipole in free space (in the absence of any dielectric/metallic interfaces) during the assumed duration of a confocal scan. See Fig. S8 of the supplementary material for the results of the calculations. The error Δα introduced by our data analysis and photon statistics is not straightforward. In order to estimate this error, we performed a bootstrapping analysis by simulating 1000 noisy intensity patterns for various orientations using the total number of photons calculated above and analyzed them using Eq. (12). We then calculate the error Δα as a function of α and z₀ from the distribution of the out-of-plane angles. Using this, we calculated the axial localization error Δα(∂z/∂α) due to the uncertainty of the orientation estimate alone. The net error $Δz|z0$ is shown in Fig. 9(a).

An additional result of the bootstrapping analysis is the lateral localization error σ_xy. Figure 9(b) shows σ_xy as a function of orientation α and axial position z₀ with respect to the surface. The number of photons for generating the intensity patterns at each axial position and orientation was taken from the calculations shown in Fig. S7(b) of the supplementary material. Assuming Poissonian statistics, σ_xy can be written as a function of the number of recorded photons using the relation $σxy∼L/N$⁠, where L is a characteristic length scale. Here, L depends on the width of a molecule’s image and is therefore a function of the dipole orientation and the structure of the electric field within the excitation focus. Figure 9(c) shows the fitted σ_xy with an average value of L ≈ 73 nm, which is in good agreement with the obtained bootstrapping result. Two important points should be highlighted here: (i) The information gained from a single photon on the lateral position of the emitter is greater if the emitter is scanned with a radially polarized focus than if it is scanned using a linearly/circularly polarized focus, where L ≈ 100 nm and (ii) as can be seen in Fig. 9(c), L is overestimated for dipole orientations close to horizontal, indicating that one has higher lateral position information per photon detected from a horizontal dipole than for a vertical dipole.

The alternative method for determining the three-dimensional orientation of an emitter is defocused imaging.^{15,16,20–23} By optimizing the same negative log likelihood function as before, Eq. (12), but this time by using the theoretical patterns for defocused images, one can repeat the error analysis of Subsection II D 2. But now, the measurement of the fluorescence lifetime and the determination of the orientation with defocused imaging are two independent experiments, which can, however, be carried out simultaneously by splitting the emission light into two different detection pathways, one focused onto a single-photon sensitive avalanche photodiode, and the other directed toward a camera (see also Sec. III). The axial localization error is again determined by the uncertainties of the lifetime and orientation estimates as given in Eq. (17). Depending on the splitting ratio of the detected photons between the lifetime and the imaging channel, these two uncertainties and their contribution to the net axial localization error will vary. For the current work, we assume a splitting of the emission in parts of 30% and 70% for lifetime estimation and defocused imaging, respectively.

Individual errors in Eq. (17) can be estimated using a similar bootstrapping approach as described in Subsection II D 2 (see also Sec. S5 B of the supplementary material). Figure 10(a) shows the axial localization error of a molecule as a function of dipole’s orientation and axial position, where we assumed that one collects 10⁴ photons from the same molecule in free space (in the absence of any dielectric/metallic interfaces). A comparison with Fig. 9 shows that the calculated axial localization errors using confocal scans with a radially polarized excitation laser and using defocused imaging in combination with smMIET are very similar.

The lateral localization accuracy σ_xy is again obtained from a bootstrapping analysis. Figure 10(b) presents this uncertainty as a function of orientation and axial position. Two contrasting differences with respect to radially polarized focus scanning can be seen: (i) the information gained from each photon on the lateral position of the emitter is much lower—if σ_xy is approximated as $L/N$⁠, then L is between 160 nm and 240 nm; (ii) L is a function of α and for dipoles oriented close to horizontal it is much larger (L ≈ 240 nm) than for vertical dipoles (L ≈ 160 nm).

In this last Theory section, we study the combination of smMIET with existing single-molecule localization based super-resolution methods such as STORM, PALM, or DNA-Points Accumulation for Imaging in Nanoscopy Topography (PAINT) microscopy for localizing single molecules with 3D nanometer resolution. The experimental setup and methodology proposed in this work will be especially useful for resolving structures of biological macromolecules, that are typically less than a few tens of nanometer in diameter, labeled with fluorophores or oligomers (when using DNA-PAINT) at multiple specific sites. In most cases, when labelling with fluorescent dyes via a flexible linker, one deals with rapidly rotating fluorophores. We consider two experimental schemes: (i) the combination of smMIET with scan-imaging using a radially polarized laser focus and (ii) the combination of smMIET with wide-field imaging using an EM-CCD camera.

Since for rapidly rotating fluorophores, the orientation effects are averaged out, the axial localization accuracy of a molecule is well captured by our theoretical results presented in Sec. II D 1. Typical image frame rates used in single molecule localization are around 100 Hz. Recent developments of high-speed-high-precision galvo-scanning systems or, alternatively, acousto-optical deflection systems make it possible to scan areas of 1 × 1 μm² with frame rates of up to 1 kHz.^24,25 A few shortcomings of this way of imaging are expensive and complex instrumental setup and low dwell time on a macromolecule’s position (a significant fraction of time the excitation focus is not positioned on the macromolecule’s location). The main advantage of using scan-imaging with a radially polarized laser is the much tighter scan focus as compared to that of a linearly/circularly polarized Gaussian beam.¹⁴ This results in higher lateral localization accuracy for the same number of detected photons. Figure 11(a) shows the result of a bootstrapping analysis of simulated intensity patterns generated by scanning a randomly oriented dipole with a radially polarized laser, as a function of the number of collected photons. The signal-to-noise ratio was set to 10. The calculated lateral localization accuracy σ_xy as a function of the number of photons was fitted using the approximation $L/N$⁠, where L ∼ 66 nm.

Let us now consider a thought experiment where an area of 1 × 1 μm² is scanned at a frame rate of 30 Hz with a scan step size of 50 nm (20 × 20 scan pixels per frame). The dwell time at each pixel is ∼80 μs. If we suppose that the laser excitation is high enough to drive the molecule, with a quantum yield of 0.6, into its excited state with probability of 0.1 per pulse (1 successful excitation event every tenth pulse), and a net collection efficiency of 5% including all the optical losses in the microscope and the metal layer, we collect approximately 20 photons in the brightest pixel with an 80 MHz laser. By considering an area under a 2D Gaussian with σ = 66/50 pixels, we would have ∼200 photons for axial and lateral localization. This would translate into an axial error of below 6 nm within the range of 20–50 nm from the MIET substrate with a 10 nm silica spacer and an accuracy of about 4.5 nm laterally.

An alternative way to perform combined smMIET-STORM/PALM/DNA-PAINT experiments is to split the photons onto a single-photon avalanche photodiode and an EM-CCD camera. Splitting the emission into parts of 30% for lifetime estimation and 70% on a camera for localization in the lateral plane allows one to illuminate the macromolecule continuously with focused laser illumination until the end of one measurement (defined by the number of image frames required to record each individual blinking molecule on the macromolecular complex at least once). Shifting the camera plane back to the conjugate focal plane of the microscope (see Sec. III for details) gives a localization accuracy that can be approximated by $100/N$ nm as determined by a bootstrapping analysis. This means that if a total sum of 1000 photons are collected within its on-time duration, resulting to 300 photons on the SPAD for lifetime estimation, and 700 photons for determining its lateral position, the molecule (with quantum yield ϕ = 0.6) can be localized with an accuracy of ∼4 nm in the xy-plane and with an axial error below ∼6 nm within the first 60 nm from the silica spacer.

Both the experimental schemes considered here yield nearly the same isotropic nanometer localization accuracy in all three dimensions. Pooling multiple localizations at the same label site (by exploiting repeated blinking in STORM, or repeated binding/unbinding in DNA-PAINT) could give sufficient statistics to measure intramolecular distances. Alternatively, data can be collected from several macromolecules and then statistically analyzed for arriving at nanometer-accurate chart of label sites (similar to methods employed in cryo-electron microscopy, see also Ref. 2). Even with the 5 nm isotropic resolution, one can achieve sub-nanometer co-localization accuracy when pooling the data from ∼100 individually measured macromolecule complexes immobilized on the surface. Moreover, the ability to determine fluorescence lifetimes of each individual blinking event can help with identifying and sorting them according to their axial positions.

For rapidly rotating fluorophores, a standard confocal microscope with Time-Correlated Single-Photon Counting (TCSPC) equipment is sufficient for localizing single molecules with nanometer accuracy in all three dimensions using smMIET imaging. In the case of experiments involving fixed dipole orientations, one has to modify the setup by introducing optical elements required for generating a radially polarized laser beam in the excitation path when using scan imaging, or to realize defocused imaging when using a camera. Figure 12 shows a schematic representation of the complete optical setup.

A white-light laser (Fianium SC400-4-80) together with an acousto-optic tunable filter (AOTFnC-400.650-TN, AAoptic, France) was used for excitation. For the experiments reported here, an excitation wavelength of λ_exc = 510 nm was picked from the white-light continuum. The spectrally filtered polarized beam was coupled into a polarization maintaining single-mode optical fibers (PMC-400-4.2-NA010-3-APC-250 V, Schäfter and Kirchhoff, Germany), and thereafter collimated into a TEM₀₀ mode with a diameter of 5 mm using an infinity-corrected 10× objective (UPLSAPO10X, Olympus). Any unwanted excitation wavelengths were additionally filtered by using a clean-up filter (FF02-510/10, Semrock). The horizontally polarized collimated light was either coupled directly into the microscope or directed through a series of optical elements that converted it to a radially polarized beam. In that case, an additional linear polarizer (LPVISE 100-A, Thorlabs, Inc.) was added to ensure high linear polarization quality of the beam. This beam was then passed through a liquid crystal mode converter (Arcoptix S.A.) which rotates the light polarization in a position-dependent manner to generate a radially polarized TEM₀₁ beam. A 25 μm pinhole was used for mode cleaning and for blocking any unwanted higher orders of vortex modes that are generated by the mode converter. This excitation beam was reflected with a non-polarizing beam splitter (ThorLabs BS019) into an objective lens with high numerical aperture (APON60XOTIRF, N.A. = 1.49 oil immersion, Olympus) which focused it onto the surface of the substrate. Scanning was done by moving the sample with a piezoelectric stage (P-562.3CD, Physical Instruments) with pixels of size 50 nm driven by using a digital piezo-controller (E-710.3CD Physik Instrumente). The excitation laser intensity was ∼4 kW/cm², and the sample scan rate was set to 3 to 6 ms per scan step. Both these parameters were chosen in such a way as to minimize photobleaching and to achieve a reasonable signal-to-noise ratio.

For recording scan images, the collected fluorescence was focused onto the active area of a single-photon avalanche photodiode (PicoQuant τ-SPAD) and counted with a multichannel picosecond event timer (HydraHarp 400, PicoQuant GmbH, Germany). The emission was filtered using a band-pass filter with a maximum transmission wavelength around 542 nm (FF01-542/27, Semrock). For defocused imaging, the collected light was split using a 70R:30T beam splitter, with 70% of light focused onto a camera and 30% onto the SPAD. The net magnification of the tube lens before the camera was chosen in such a way so that the size of one pixel in sample space is approximately 100 nm. The camera was shifted along the optical so far as required for generating a defocusing of z′ = 0.6 μm in sample space. An optional glass slab can be inserted into the optical path between the tube lens and the camera as shown in the figure. This allows for increasing the optical path length and to switch between focused and defocused imaging. Measurements were done in the following way: first, confocal scan images were acquired to determine the position of the individual molecules/labeled complexes. Subsequently, point measurements were performed by parking the excitation laser at the identified positions and placing the beam splitter between the SPAD and the camera. Defocused images and lifetime data were collected simultaneously for each point till photobleaching. All data collection and hardware synchronization were performed using LabVIEW.

Glass coverslips (thickness 170 μm, refractive index 1.52) were cleaned with piranha solution (3:1 v/v ratio of concentrated H₂SO₄ and 30% w/v H₂O₂) for about 15 min. These were later washed with water and used as substrates for vapor deposition of 2 nm titanium, 10 nm gold, 1 nm titanium, and a SiO₂ spacer with the required thickness. The process was carried out under high-vacuum conditions (≈10⁻⁶ mbar) by using an electron beam source (Univex 350, Leybold). A slow rate of deposition was of (1 Å s⁻¹) was used to ensure maximum flatness of the surface. Layer thickness was monitored by using an oscillating quartz unit during deposition and was later verified by atomic force microscopy. 10 μl of 0.1 nM Atto 655 or (0.1% w/v) PVA/water solution, with 0.1 nM rhodamine 6G molecules, were spin-coated on top of the substrate at ∼8000 rpm for 60 s.

We have developed a dedicated Matlab-based Metal-Induced Energy Transfer Graphical User Interface (MIET-GUI) for evaluating measured smMIET data. The software can be downloaded at https://projects.gwdg.de/projects/miet/repository. The MIET-GUI accepts .ht3- and .ptu-files generated by the TCSPC hardware HydraHarp (PicoQuant GmbH, Berlin), from which it calculates the fluorescence lifetime and intensity scan images. The lifetimes are converted into height values via the pre-calculated lifetime-versus-height calibration curves. The theoretical calculations of the intensity patterns that are generated by scanning the molecules with a radially polarized laser, the pattern matching, and the calculation of the lifetime-versus-height curves are performed by the GUI using all required sample properties as specified by the user. Next, the axial positions of the molecules are calculated by combining the lifetime and orientation information. Finally, the axial position error can also be calculated using the “Bootstrap” option in the GUI that estimates this error as described in detail in Sec. II D.

In our earlier publication,⁹ we measured Atto 655 molecules that were spin-coated on top of MIET substrates with various thickness values of the silica spacer evaporated on top. The substrates were scanned with a standard confocal microscope, and lifetime images were recorded. The lifetime values measured from approximately 400 molecules on top of 20 nm, 30 nm, 40 nm, and 50 nm silica spacer were (0.50 ± 0.06) ns, (0.81 ± 0.07) ns, (1.19 ± 0.08) ns, and (1.50 ± 0.08) ns, respectively. On average, N = 370, 770, 1000, and 1050 photons were recorded per molecule, respectively. The lifetime values were in excellent agreement with the theoretical model (fits are shown in Fig. 4 in Ref. 9). The results showed that we consistently achieved an axial localization accuracy of ∼2.5 nm for all spacer thickness values. Figure 13 shows the theoretical lower bounds for axial errors for various numbers of photons collected from a dye molecule placed directly on top of a silica spacer of thickness z₀. These calculations show that one can indeed localize single molecules with <2 nm accuracy along the axial direction using as few as 450 photons across a distance range of 20 nm–60 nm. The axial localization error increases for spacer thicknesses less than z₀ = 5 nm due to the complete quenching of fluorescence. The lifetime estimation errors from the experiments were slightly larger than $τf/N$⁠. This can be due to the presence of background that can perturb the fitted lifetime values of individual molecules.²⁶ The localization accuracy estimated using Eq. (15) for the measured lifetime uncertainties are 2.3 nm, 2.1 nm, 2.2 nm, and 1.9 nm, respectively.

The new result in the present paper is the determination of the lateral localization accuracy of these scanned individual molecules. Let us consider a scan image, shown in Fig. 14(a), acquired with Atto 655 molecules spin coated on top of a substrate with a 40 nm silica spacer. The scan step size and dwell time are 67 nm and 6 ms, respectively. On average, 1370 photons were collected from each emitter in this image. The standard deviation of the PSF estimated from the intensity scans of molecules by fitting a 2D Gaussian distribution is around 100 nm, and the background noise was estimated as 27 photons per scan step. This translates into an average single molecule lateral localization accuracy σ_xy of about 16 nm [Eq. (13)]. Photoluminescence of the thin gold film due to weak d-d transitions is the biggest source of background in our confocal scan images. The lifetime of this spectrally broad photoluminescence is on the order of a few picoseconds. Recording photons in TCSPC mode gives us the unique advantage of rejecting this gold photoluminescence by time gating. For this purpose, we retain only photons which arrive after a minimum delay of ∼1 ns after the preceding laser pulse, as shown in the inset of Fig. 14(d). This improves the signal-to-noise ratio by a factor of ∼4 as shown by the intensity line profile in Fig. 14(d). As a net result, the lateral localization uncertainty is reduced by approximately the same factor, and the average single molecule localization accuracy σ_xy is around 4.5 nm [see Fig. 14(e)]. This result is consistent with the fact that the localization uncertainty can be seen as a sum of two terms: one term, which is inversely proportional to the square root of the number of photons, and a second term which is inversely proportional to the signal-to-noise ratio N/B, see Eq. (14). For a high signal-to-noise ratio, the lateral localization accuracy follows the inverse square root rule. This shows that we indeed localized each individual emitter with an uncertainty of ∼2.5 nm axially and ∼4.5 nm laterally.

Figure 15(a) shows an exemplary scan image of Rhodamine 6G molecules embedded in a thin PVA polymer. This scan was performed with a step size of 50 nm and a dwell time of 5 ms. Time-gating photons with a gate-opening time of ∼1 ns show considerable improvement in signal-to-noise ratio, as shown by the intensity plots along a line, see Fig. 15(e). With the chosen intensity and scan rate, the total number of photons collected from each molecule ranged from 1 × 10³ to 8 × 10³. For data evaluation, exact wave-optical calculations were performed to compute a discrete set of theoretical excitation scan images^27,28 for various azimuthal (β) and polar (α) angles (see Fig. S5 of the supplementary material). These master patterns were used to fit the intensity images using a least-squares minimizing pattern matching algorithm.¹⁶ The obtained fit parameters served as initial guess values for the optimization of the Poisson-noise based negative log likelihood function [see Eq. (12)] which yields refined parameters beyond the discrete set of values recovered by the pattern matching. The fitted patterns for the measured intensity images in Figs. 15(a) and 15(b) are shown in Figs. 15(d) and 15(e), respectively. TCSPC histograms of each individual emitter were constructed from all photons belonging to that emitter. These histograms were fitted with an exponential decay model to determine their fluorescence lifetimes. In this way, the orientation and the lifetime of each emitter were estimated together with their uncertainties, which were used to determine their axial positions. Figure 15(f) shows a false color image of the distances of the molecules from the silica spacer.

To estimate the fitting errors, we applied bootstrapping as already discussed in Sec. II. A distribution of each parameter was obtained by fitting thousand re-sampled images using the maximum likelihood estimator. Figure 16 shows the result for four Rhodamine 6G molecules immobilized in a PVA thin film. About 1800 photons were recorded from molecule #1, which was embedded almost parallel to the substrate. Its lateral position was determined with an uncertainty of ∼2.5 nm. The axial position and the uncertainty related to it were determined by using the estimated distribution of the dipole’s orientation and the fluorescence lifetime, which is about 1 nm. The axial position of molecule #2 is ∼10 nm above the substrate in the polymer. The excitation dipole of this molecule is oriented at (27 ± 6)° with respect to the optical axis. Using the error Δα, the axial localization uncertainty of molecule #2 was calculated to be ∼0.4 nm albeit the large uncertainty in determining its orientation. This can be explained with the help of the curves showing the fluorescence lifetime as a function of height and inclination (Fig. 6). Close to z₀ = 10.5 nm, the MIET lifetime-versus-height curves intersect each other which leads to ∂τ/∂α ≈ 0. In this case, Δz(z₀ = 10.5 nm) is dominated by Δτ_f. However, the larger uncertainty in estimating the orientation of the dipole and the presence of additional molecules in the neighborhood that contribute to the background parameter lead to an increased error in the estimation of its lateral position, which is about 8.5 nm.

The results of about 400 measured molecules are shown in Fig. 17. In Fig. 17(a), σ_xy of the molecules is plotted as a function of the number of photons, and in Fig. 17(b), the distribution of σ_xy versus Δz determined for each molecule is shown. Two important points should be mentioned here. First, the lateral localization errors are in excellent agreement with the simulation results obtained in Sec. II D 2, i.e., $σxy≥L/N$ with L ≈ 73 nm. Second, the accuracy of determining the axial position of the molecules, Δz, is better than σ_xy. This can be explained as follows: For a molecule whose lifetime is modulated by the energy transfer to the metal surface, the arrival times of each detected photon contain information about its axial position only within the first few tens of nanometers (MIET range). If we assume that the lifetime modulation is almost linear within distance z_range, the axial localization error can be approximated as

For example, for a dipole with an inclination of ∼50°, one has $Δz≈20τf(z0)/N$ nm/ns. Now, assuming σ_xy ≈ 100 nm, this means that one has 3 to 4 times higher localization accuracy along the axial direction than along the lateral direction. The results tabulated in the caption of Fig. 16, and the distribution shown in Fig. 17, are in good agreement with this approximation. These results demonstrate that we determined the position of individual fixed dipole emitters with nanometer accuracy in all three dimensions.

Equation (18) holds true for freely rotating dipoles as well. By inspecting Fig. 2, one can say that irrespective of the quantum yield, the axial resolution can be written as $Δz≈100(τf/τ∞)/N$ nm. The accuracy becomes better the closer the emitter is to the metal surface, and it can be up to 5 times larger than the lateral localization accuracy.

In this work, we have developed a complete theoretical framework for smMIET experiments in conjunction with orientation and orientation imaging (by either scanning or wide-field imaging) to localize individual fluorescent emitters with nanometer accuracy along all three spatial dimensions. Furthermore, we have presented first experimental results for sparsely distributed molecules. When combining smMIET with techniques such as STORM, PALM, PAINT, or potentially the recently introduced MinFlux,²⁴ it can provide super-resolution microscopy in all three dimensions with nanometer accuracy, within a range of ∼150 nm above a surface. This opens the possibility to use it for resolving structural details in single macromolecules or macromolecular complexes that are suitably labeled with fluorescent labels. Finally, in most cases, labeling is done via flexible linkers allowing labels to rotate freely. This tremendously simplifies smMIET because it requires only a conventional confocal microscope with fluorescence-lifetime measurement capability. Thus, we hope that smMIET imaging will find broad applications in all fields of research, from biology over chemistry to physics.

See supplementary material for the complete derivations of electric field, angular distribution of radiation, and collection efficiency function for a dipole on top of an interface. A brief introduction to scanning with a radially polarized laser and defocused imaging is also given. Detailed calculations for axial localization errors using both the methods are available here.

The authors thank Sebastian Isbaner for critically reading the manuscript and for his valuable insights. Funding from the Deutsche Forschungsgemeinschaft is gratefully acknowledged (SFB 860, Project No. A6; SFB 937, Project Nos. A5 and A14). N.K. and J.E. thank the DFG Cluster of Excellence “Center for Nanoscale Microscopy and Molecular Physiology of the Brain (CNMPB)” for financial support.