Defining a New Diffraction Limit
Super Resolving Power Using Feynman’s Resolution Rule
- Fig. 1: The resolving power of an optical system.
- Fig. 2: State percentage of population’s ratios of FM versus logarithmic normalized laser power. A and B refer to fluorescence’s population ratio (NF/NT) mesured from both images shown in figure 3.
- Fig. 3: (A) STEF effect: fluoresence intensity under high laser power pulse (100% FM are fluorescing) and (B) maximum intensity measured from the same specimen under continuous laser power (less than 15% of FM fluoresce). Scale bar is 10 µm.
Super-resolution techniques outperform Abbe’s diffraction limit, urging for a new theoretical framework. Prior to the super resolution’s era, Feynman used a more general definition: instead of using the minimum distance to distinguish two points, he employed the minimal time. We use Feynman’s resolution rule to show a relation between PALM and STED techniques. Finally, experimental validation of this relation led us to discover a new process we called STimulated Emission Fluorescence (STEF).
To date, the empiric resolution obtained from Super Resolution (SR) techniques cannot be explained by Abbe’s diffraction limit, highlighting the necessity to develop a new theoretical framework. Interestingly, before SR demonstrations, Richard Feynman had defined the resolving power in more general terms than Abbe. In order to explain theoretically how resolution can be improved to reach SR, we adapt Feynman’s resolution rule to fluorescence and explain Photo-Activated Localization Microscopy (PALM). The fluorescent process and stimulated emission is later detailed, demonstrating further a relation between PALM and STimulated Emission Depletion (STED) resolution expressions supported by experimental measurements. Furthermore, this experiment brought to light a new phenomenon called the STimulated Emission Fluorescence (STEF).
The New Definition For Resolution
Contrary to Abbe’s definition of resolution based on the minimal distance between two distinct objects, Feynman  refers to temporal expression of the resolution :
“For a particular point P all the rays from object to image T take exactly the same time (fig. 1). The condition that the second point is focused in a distinctly different place from the first one is that the two times for the extreme rays P′ST and P′RT on each side of the big opening of the lenses to go from one end to the other, must not be equal.
The general rule for the resolution of any optical instrument is this: two different point sources can be resolved only if one source is focused at such a point that the times for the maximal rays from the other source to reach that point, as compared with its own true image point, differ by more than one period.
where ν is the frequency of the light.
If the distance of separation of the two points is called D, and if the opening angle of the lens is called θ, then one can demonstrate that eq. 1 is exactly equivalent to the statement
where n is the index of refraction at P and λ is the wavelength.”
This last statement is Abbe’s diffraction limit and is obtained from Feynman’s diffraction limit definition assuming the travel interval times t1 and t2 have their onset time t10 and t20 at the exact same time. SR gain can be deducted from this statement if adapted to the fluctuating onset time encountered with fluorescent molecules (FM).
Photo-Activated Localization Microscopy (PALM)
When using FM and controlling the onset time t10 and t20 of two photon’s emission, the proximity of two FM can therefore reach a smaller distance than D to be distinguished according to Feynman. When measuring one photon after the other from a single FM, the precision of the location increases with the square root of photon measured number (√NP) from this single molecule . This is single fluorescence imaging. Photo-Activated Localization Microscopy  (PALM) uses the triggering of the fluorescence process to obtain the precise location of a subset of FM. This allows the measurement of events within a precision DPALM = D/√NP and by cumulating many subsets, the super resolved image is obtained. When considering a number of total distinct FM (NT) within a circular area of diameter D (AD) the average distance between these molecules is DSR = D/√NT. This means that measuring NP times one photon from one molecule is equivalent to count NT distinct molecules under AD at the same time.
Fluorescence and Stimulated Emission
The fluorescence process begins with the photon absorption. The excitation photon transfers its energy to one electron of the FM. The probability of absorption depends on the cross-section (σ) of the FM. The electron of the FM is accelerated into an excited state. This electron will dissipate some of its energy through radiation processes and will no longer stay into the excited state. This excited state last typically 10-9 seconds and is characterized by the lifetime decay constant (κ) of the FM. The electron then decelerates and moves back to ground state. This movement produces a new photon; the emission photon which has a different wavelength than the excitation photon. Stimulated emission differs from the fluorescence as the deceleration which returns the electron back to ground state is stimulated by the passage of another photon. The emitted photon has the same wavelength than the photon used to stimulate the return to ground state. The fluorescence process is shortcutted and the emitted photon is not distinguishable from the excitation.
STimulated Emission Depletion (STED)
STED  requires higher laser power than saturation power. Saturation power (ISAT) occurs when the photon absorption rate is equal to the spontaneous fluorescent photon's emission rate or ISAT =κ⁄σ. The Beer-Lambert law gives that from a number of excited FM (NE), NSTED molecules will return to ground state via stimulated emission under laser power ISTED:
The FM number going back to ground state via fluorescence emission (NF), is not depending on the laser power:
Therefore, when increasing the laser power higher than ISAT the FM number NF decreases while NE still increases. The ratio NE over NF is equivalent of distinguishing NT molecules simultaneously under AD:
Equation 5 allows us to write the equivalence between PALM and STED super-resolution distance DSR as:
Figure 2 shows the different theoretical population’s state ratio in function of the logarithmic normalized laser power excitation. The ratio from eq.5 is the ratio of the dotted blue curve over the solid red curve.
Materials and Method
Paraformaldehyde-fixed HEK 293T cells overexpressing cytoplasmic green fluorescent protein (GFP) were labelled using rabbit anti-GFP antibody and goat anti-rabbit secondary antibody coupled to AF-647 and put in PBS-Cysteamine buffer solution. Specimen was observed under a Zeiss Elyra system for time-lapse of 100 ms image acquisition at 1s frame rate with logarithmic increase of excitation laser power.
Under a continuous laser excitation, we measured the ratio change of the fluorescent population when increasing laser power (fig. 2). At low power, the fluorescence (solid red line) linearly increases until it reaches the saturation power (vertical gray line). Then, it starts to decrease since STED becomes more important.
STimulated Emission Fluorescence (STEF)
We are still studying another phenomenon which we attribute to stimulated emission fluorescence, i.e. that all the excited probes emit a fluorescent photon and no molecule lose their photon from the stimulated laser photon. This can only happen if the laser excitation is stopped while most of the FM are still in an excited state. Figure 3 shows the difference in intensity captured when using a single pulsed excitation (A) while the same specimen (B) shows the maximum fluorescence intensity obtained under continuous laser excitation.
The ambiguity related to breaking the diffraction limit with super resolution techniques is now elucidated via Feynman’s resolution expression. We also have shown the equivalence between PALM and STED expressions. This theoretical research brought an unpredicted phenomenon which needs more investigation: the STimulated Emission Fluorescence (STEF).
Monique Vasseur from Photonic Microscopy Plateform UdeM for usage of the Zeiss Elrya system and Michael Housset for specimen preparation and writing improvement.
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