The noise-limited-resolution for stimulated emission depletion microscopy

With recent developments in microscopy, such as stimulated emission depletion (STED) microscopy, far-field imaging at resolutions better than the diffraction limit is now a commercially available technique. Here, we show that, in the special case of a diffusive regime, the noise-limited resolution of STED imaging is independent of the saturation intensity of the fluorescent label. Thermal motion limits the signal integration time, which, for a given excited-state lifetime, limits the total number of photons available for detection.

This leads to the question of what might be the resolution limit of these new imaging techniques.However, research has focused on how the resolution scales with STED laser intensity.[4] To our knowledge no previous work has investigated under what conditions the resolution scaling law might fail.Although the diffraction limit no longer applies, fundamental sources of noise, such as shot noise and thermal noise must, at some point, play a role in limiting the resolution of all sub-diffraction-limited resolution microscopy techniques.
In this paper, we analyse the behaviour of the signal-to-noise ratio of STED microscopy.In particular we examine the limits of thermal noise (in the form of Brownian motion of the sample), detector shot noise, and sample damage threshold on resolution.The shot noise limit is found to be strongly dependent on the saturation intensity of the emitter, while thermal noise limits the signal integration time to a few milliseconds.Taking into account the optical damage threshold of the sample, we show that there exists an optimum emitter saturation intensity that gives the best resolution.

Theory
We use signal-to-noise ratio (R) considerations to determine what the resolution limit of STED is.We consider that the noise-limited resolution is given by the minimum volume that yields R ≥ 1.Assuming low light-level conditions, where shot-noise dominates, and neglecting sources of technical noise, the signal-to-noise ratio of the detector current, i, is given by R = i/ īrms .The photo-induced current in the detector is given by (1) i = Q e Φ N p T where Q e is the carrier charge, Φ is the efficiency of the optical train and detector process (e.g., the fraction of emitted photons that result in an electron), while Np T is the number of photons per unit time.The shot noise of the detector is where τ o is the time over which the measurement is performed, also referred to as the integration time.From equation ( 2), R can be increased to arbitrarily large values given sufficient observation time.However, there are practical and fundamental limits to this approach.For biological samples, the label molecules will be subject to Brownian motion and this will limit the observation time, τ o .
To calculate this time, the volume from which the signal is collected must be calculated.
The diffraction limit is given by which, for STED microscopy is modified by the STED laser to become (4) where λ is the wavelength of the excitation laser, NA is the numerical aperture of the objective lens, I is the intensity of the STED laser, and I sat is the saturation intensity of the label molecule.The label movement must be smaller than d S during the observation time, which provides the upper limit to the observation time (5) τ o = λ where T f is the temperature of the fluid, k b is Boltzmann's constant, r is the radius of the emitter, and η is the dynamic viscosity of the fluid.Clearly, as the resolution increases, the observation time must decrease commensurately, providing an upper limit to R.
The number of photons emitted from the volume defined by the node of the STED beam per unit time is given by the number of emitters (ρ e V 0 , where ρ e is the density of emitters) in the volume V 0 = π 4 d 3 .Each emitter can only emit a single photon at a time and spends on average τ time in the excited state.Thus, in observation time τ o each emitter can be expected to provide τo τ photons.Using equation ( 3) and ( 5), we can write (6) N p T = V 0 ρ e τ I sat I substituting equation ( 6) into equations ( 1) and (2) leads to the following expression for R while in the absence of Brownian motion, observation can, in principle, continue indefinitely, leading to τ I As a result, the best possible resolution for STED microscopy on liquids is given by ( 9) For solids, the resolution is limited to Equations ( 9) and (10) are the main results of this paper.In the following, we investigate the consequences of this finding.Note also that in equation( 10), the resolution can be arbitrarily increased by simply increasing the observation time.However, such a scenario is unrealistic since sources of technical noise become increasingly important.
Returning to the signal-to-noise ratio, note that R decreases with increasing STED laser intensity as expected, since the number of emitters in the volume decreases.Furthermore, R also increases as the excited state lifetime decreases because the number of photons per emitter per unit time increases.However, I sat and τ are not independent quantities with (11) where σ(ω) is the stimulated emission cross section and ω is the frequency of the STED laser.To increase the resolution of the STED process, while keeping a low STED laser intensity, it is desirable to find labels that have a low saturation intensity.However, a reduced I sat is often accompanied by an increase in excited state lifetime.According to equation ( 7), R will decrease with both a decrease in I sat and an increase in τ , thus, reducing the saturation intensity of the label will not necessarily lead to an increase in resolution.

Results
The effect of requiring R ≥ 1 on the resolution and required signal acquisition time is examined with the following parameters kept fixed: NA = 1.3, ρ e = 10 15 cm −3 , λ = 800 nm, and Φ = 1.
Fig. 1 shows how R varies with resolution and the required STED laser intensity as a function of resolution.For a solid sample (negligible Brownian motion), the saturation intensity and excited state lifetime of nitrogen vacancy centres in diamond are used (I sat = 6.6 MW.cm −2 , τ = 11 ns) [3].We also choose a signal integration time that is the maximum possible at video frame rates (33 ms), which results in a maximum achievable resolution of 0.42 nm.Unfortunately, this resolution requires a STED laser intensity of about 550 GW.cm −2 , which is unlikely to be withstood without damage to the sample.Note that at a resolution of 6 nm, the STED laser intensity is ∼2.5 GW.cm −2 , close to the value reported by Rittweger et al [3].Our agreement with experimental values is achieved by fitting a single free parameter (not included in the analysis above), called the pattern steepness factor [4].Once set, the pattern steepness is held constant for all other calculations.
For liquid samples (non-negligible Brownian noise) with a label saturation intensity of 11 MW.cm −2 and an excited state lifetime of 11 ns [4] has a lower achievable resolution at 3.6 nm.In this example, the emitter is considered to have a 20 nm radius immersed in water with a temperature of 300 K.The Brownian motion limits the observation time to less than 5 ms.
Finally, it is quite clear that such resolutions cannot be achieved using current fluorescent molecules and colour centres because the required STED laser intensities are too high.To obtain these resolutions, the saturation intensity of the emitter needs to be lower.However, molecular fluorophores with low saturation intensities usually also have a longer excited state lifetime, rather than a significantly increased stimulated emission cross section.In what follows, we have modelled this by taking I sat = A/τ with A set such that τ = 11 ns gives I sat = 11 MW.cm −2 .As shown in Fig. 2, to achieve sub-nanometer resolution and keep the STED laser intensity relatively low (∼GW.cm−2 ), one requires that the saturation intensity of between 1 and 10 kW.cm −2 .
If the signal-to-noise ratio and optical damage thresholds are considered, then a different picture emerges.Fig. 3 shows the signal-to-noise ratio as a function of saturation intensity and resolution.The white area in both subfigures indicates parameters that either yield R < 1 or a STED laser intensity greater than 100 GW.cm −2 , which is an optimistic assessment of the optical damage threshold.Fig. 3(a), shows the results for solids with an observation time of 33 ms, while (b) shows the results for liquids using the same liquid parameters as for Fig. 1. Surprisingly, the highest resolution is not obtained with the lowest saturation intensities because the long excited state lifetime results in too few photons and the signal-to-noise ratio is too low.Instead, the minimum resolution is 0.1 nm at a saturation intensity of 50 kW.cm−2 for solids and 0.4 nm at a saturation intensity of 1 MW.cm −2 for liquids.Note that the presence of Brownian motion forces one to use labels with a shorter excited state lifetime in order to gather sufficient signal before the emitter diffuses out of the central node of the STED laser beam.This reduces the attainable resolution significantly.The maximum resolution for both solid and liquid conditions is smaller than the radius of the emitter (20 nm for the liquid sample).In fact, for large fluorescent molecules, such as green fluorescent protein (r ≈2.82 nm [5]), it is questionable if resolution below the size of the emitter has physical meaning.For instance should  excitation and emission occur at different sites of the same molecule, as is true for green fluorescent protein (see [6] and references therein), then the STED process would be disrupted.Thus, in these extreme cases, it is likely that the minimum resolution is given by the separation between the absorption and emission locations on the same emitter.
Fig. 3 also indicates that for a sample with a particular damage threshold there is an optimum saturation intensity and a resulting minimum resolution.This relationship is clarified in Fig. 4, where the resolution is plotted as a function of the optical damage threshold and the saturation intensity.The colour scale of Fig. 4 has been cut off so that the region of optimum resolution has an expanded scale.Note that for robust materials, a wider range of saturation intensities will result in nanometer resolution than for low damage threshold materials.This indicates that high resolution STED microscopy on low damage threshold materials will require a careful choice of label.Furthermore, under these conditions, the sample will probably have to immobilised and long observation times used to acquire the image.
From our calculations, for STED microscopy with emitters attached to a molecule with a hydrodynamic radius less than 20 nm, the signal integration times are always limited by Brownian motion to less than 5 ms (see Fig. 5), which, in turn, requires labels with a short excited state lifetime to achieve R >1.Since labels with a short excited state lifetime typically exhibit higher saturation intensities, the required STED laser intensity is increased.This is undesirable, since it brings the STED laser intensity closer to the damage threshold of the sample.

Conclusion
These calculations show that, although the diffraction limit does not limit the resolution of STED microscopy, its resolution is limited by shot noise, thermal noise and damage threshold considerations.The exact resolution limit depends on the exact emitter properties, however, using typical values of current labels, we have shown that this is 0.42 nm for solids and 3.6 nm for liquids.Although our calculations have focused on fundamental noise, technical noise should be considered as well.For instance, at a resolution of 6 nm and centroid positional accuracy of 0.1 nm, STED technology has already reached the point where instrumental concerns, such as microscope stage drift must be taken into account.With drifts commonly on the order of 0.1 nm.s −1 , [9] any dynamics extracted from such high resolution images must account for and remove such systemic noise.
On the other hand, these calculations also show that STED has yet to reach its full potential and the development of new labels with appropriate combinations of saturation intensities and excited state lifetimes should result in a considerable improvement in resolution over existing results.Moreover to achieve sub-nanometer resolution, labels with a saturation intensity of the order of ∼100 kW.cm −2 and excited state lifetimes of the order of nanoseconds must be developed.This combination cannot usually be met by dye molecules, but may be achievable in quantum dot emitters, where it appears that low saturation intensities may be combined with short excited state lifetimes.[7,8].

2 )Figure 1 .
Figure 1.An example of the signal-to-noise ratio (solid lines) and required STED laser intensity (dotted lines) as a function of resolution for a liquid (grey lines) and a solid-state (black lines) sample.Dashed lines indicate the conditions for which R = 1.

2 )Figure 2 .Figure 3 .
Figure 2. Required STED laser intensity as a function of saturation intensity and resolution.

Figure 4 .
Figure 4. Resolution (nm) as a function of optical damage threshold and saturation intensity.

Figure 5 .
Figure 5. Signal integration time as a function of desired resolution and the radius of the molecule the emitter is attached to.The liquid parameters are the same as described in Fig. ??.The saturation intensity of the label was assumed to be 11 MW.cm −2 and the excited state lifetime 11 ns.