Calculating the Cramer-Rao Lower Bound (CRLB)

The CRLB is a best-case estimator of the precision of the fitting parameters. We use it to calculate the theoretical precision of dSALM and vSALM (Fig. 1e), and to assign experimental localization precisions to every single molecule. The CRLB can be calculated from the inverse of the Fisher Information matrix FIu,v^19,33:

with

μ_k(θ_u) is the model describing the intensity in each pixel k. In our case, it is an experimentally derived spline-interpolated PSF model. θ_u = {x, y, z_as,N, b} are the fitting parameters that include the position of the fluorophore x, y, z_as, the number of photons N and the background per pixel b.

For dSALM, we estimate the lateral localization precisions δx=CRLBx and δy=CRLBy and the axial PSF-based localization precision δzas=CRLBz,as from the CRLB of the UAF channel only. The photometry-based axial localization precision in dSALM we calculate from the precision δNS=CRLBN,S and δNU=CRLBN,U of the number of photons detected in each channel using Eq. ⁵. To this end, we start with Eq. ³, use the definition f(z) = N_S/N_U and apply Gaussian error propagation¹⁹:

The combined axial localization precision from photometry δzdSALM and PSF shape δzas we calculate as the weighted average of each localization precision (compare with Eq. ¹³ in the Data analysis section):

For vSALM, we estimated the lateral localization precision as the weighted average of the localization precisions of the UAF and total fluorescence channel, e.g.,:

The axial localization precision based on photometry we calculate analogous to the dSALM example but with N_S(z) replaced by N_Tot(z) and fv(z)=ITot(z)/IU=f(z)+1:

If we approximate δN with N, we can directly see where the improvement of dSALM vs vSALM comes from:

First, in vSALM the beam splitting leads to a decrease in NU by a factor of 2 compared to dSALM. Second, especially for large z, NTot≫NS. Both increase δf_v compared to δf.

For calculating the CRLB in Fig. 1e, we assumed that we collect N_c = 5000 photons from a fluorophore far away from the coverslip. For fluorophores with a low quantum yield (as is the case for our experimental data with a quantum yield on the order of 30%) SAF competes not only with UAF, but mostly also with the nonradiative decay. Thus, we made the approximation N_U = N_c and N_S(z) = f(z)N_c. For vSALM, after beam splitting N_U = N_c/2 and N_Tot(z) = (N_U + N_S(z))/2. For fluorophores with high quantum yield, SAF competes with UAF and N_U + N_S = N_c. Then NU(z)=Nc/(1+f(z)) and NS(z)=Ncf(z)/(1+f(z)). The curves in Fig. 1e, although different in details, remain qualitatively the same.