Encoding and decoding spatio-temporal information for super-resolution microscopy

The challenge of increasing the spatial resolution of an optical microscope beyond the diffraction limit can be reduced to a spectroscopy task by proper manipulation of the molecular states. The nanoscale spatial distribution of the molecules inside the detection volume of a scanning microscope can be encoded within the fluorescence dynamics and decoded by resolving the signal into its dynamics components. Here we present a robust and general method to decode this information using phasor analysis. As an example of the application of this method, we optically generate spatially controlled gradients in the fluorescence lifetime by stimulated emission. Spatial resolution can be increased indefinitely by increasing the number of resolved dynamics components up to a maximum determined by the amount of noise. We demonstrate that the proposed method provides nanoscale imaging of subcellular structures, opening new routes in super-resolution microscopy based on the encoding/decoding of spatial information through manipulation of molecular dynamics.

the following parameters: confocal FWHM=200 nm, particles distance = 104 nm,  0 = 4 ns, k S = 1, S = 10 12 , B = 0, T = 12.5ns. In this particular example the gated image with maximum resolution is obtained by setting T g = T = 12.5 ns. On the contrary, the resolution of the SPLIT image can be increased by using a higher number n of components (shown is n = 6). This example is provided to illustrate that time-gating and SPLIT operate in a different way and doesn't correspond necessarily to a practical case. The colormap represents the simulated intensity normalized to the maximum value of each image. Scale bar 100 nm.

Modeling of the STED decay components
For simplicity of calculation, we assume a 3D Gaussian profile of the confocal PSF, with waists along the x, y and z directions given by w x =w y =w and w z respectively: where we have defined the radial part of the confocal PSF as: Then we approximate the doughnut-shaped intensity distribution of the STED beam at the focus as a parabolic function of the radius and ignore for simplicity any dependence along z: The instantaneous probability of stimulated emission depends linearly on the STED beam intensity: where  0 =1/ 0 is the decay rate of the spontaneous emission and the constant I SAT is usually called saturation intensity 1 and represents the value of intensity for which  STED = 0 .
The resulting decay rate as a function of the position is: The time-dependent fluorescence intensity F(x,y,t) at each pixel can be expressed as: where K is a constant that depends on the quantum yield of the fluorophore, the maximum of the excitation intensity and the detection efficiency, r 2 = (x'-x) 2 + (y'-y) 2 and (x',y',z') is the density of fluorophores. We conveniently switch to a system of cylindrical coordinates centered on the pixel (x,y) and integrate along z' and ':

Discretization of the continuous distribution of STED decay components
In order to approximate the continuous distribution of decays in a discrete number n of components, we split the integral into n parts: ... , , We need to get n components which do not depend on the function C(r 2 ). For this reason we expand C(r 2 ) as a Fourier series inside the interval (r i-1 2 , r i 2 ) of width r i 2 : and approximate the function C(r 2 ) with the term of order zero:   We are approximating C(r 2 ) inside (r i-1 2 , r i 2 ) with its average value within this interval, and ignoring its variations inside (r i-1 2 , r i 2 ).
We define the time-dependent decay of the i th component as: A smaller value of the ratio r 1 /w is associated to a resolution improvement of the first image of the SPLIT series with respect to the confocal image. The FWHM of the effective PSF of the SPLIT image decreases with increasing values of n as shown in Supplementary Fig. 2 for the values k S = 10, k S = 1 and k S = 0.1.
If we substitute the function C(r 2 ) with a constant C, the analytical form of the decay is given by: which can be used to extract the parameter k S from the experimental data.

Supplementary Note 2 -Explicit expression of the SPLIT image
The first component of the SPLIT image (n=2) can be explicitly expressed as a function of the phasor components: Where N(x,y) is the time-integrated intensity and M -1 is the inverse of the matrix M. The images of the phasor components g(x,y) and s(x,y), extracted from the temporal dynamics, potentially encode the additional spatial information. This extra amount of information is evident for instance in the case of a CW-STED image whereas is absent in a confocal image (see Supplementary Fig. 3 where the images of  phase (x,y)=(T/2)(s(x,y)/g(x,y)), a parameter independent from the amount of background, are reported).
We can explicitly express the above equation as a function of the time-resolved fluorescence intensity and obtain: CW-STED, for low values of k S the separation into only n=2 components may not lead to significant reduction of the SPLIT E-PSF. If this is the case, further improvement in resolution can be obtained by separation into a larger number n of components, corresponding to an effective E-PSF n SPLIT containing additional terms which extract spatial information from higher harmonic content of the temporal dynamics.

Supplementary Note 3 -Propagation of the noise in the SPLIT method
The SPLIT image, N i (x,y)=f i (x,y)N(x,y), is affected by additional noise brought in by the factor f i (x,y). The fraction f i is calculated at each pixel from the measurement of g and s at one or multiple harmonics. Here we discuss how the noise in the measurement of g and s (or higher harmonics components) is propagated to the fractions f i in the linear system Mf P  depending on the properties of the matrix M. The error propagation through a linear system is usually quantified by considering the 'condition number' k cond defined as the product of the norms of the matrices M and its inverse M -1 : The parameter k cond is dependent on the particular choice of the n vectors (g 1 , s 1 ,…), … , (g n , s n ,…) which form the matrix M and which describe the n dynamics components. This parameter decreases with higher values of k S and increases with the value of n. This dependence is shown for the values k S = 1 and k S = 10 in Supplementary Fig. 4. It can be seen from the figure that it is possible, in principle, to get with the value k S =1 (n = 4) a resolution comparable to that obtained with k S = 10 (n=2) but with a noise propagation which is 2 orders of magnitude larger. Because