Fig. 1. Fundamental data model example. (a) Photon distribution profile $f_{\theta }$ for an in-focus and stationary molecule whose image is modeled with the Airy pattern. This profile assumes that the molecule is located at $(x_0,y_0)=(0\mathrm{nm},0\mathrm{nm})$ in the object space and that photons of wavelength $\lambda =520\mathrm{nm}$ are collected using an objective lens with a numerical aperture of $n_a=1.4$ and a magnification of $M=100$ . While the profile is defined over the detector plane ${\mathbb{R}}^2$ , the plot shows its values over the $150\mathrm{\mu m}\times 150\mathrm{\mu m}$ region over which it is centered. (b) A fundamental image simulated according to the profile in (a), with each dot representing the location at which a photon is detected. The simulation assumes a constant photon detection rate of ${\mathrm{\Lambda }}_0=10,000\text{photons}/\mathrm{s}$ and an acquisition time of $t-t_0=0.05\mathrm{s}$ , such that the mean photon count in an image is ${\mathrm{\Lambda }}_0·(t-t_0)=500$ . In this particular realization, $N_0=516$ photons are detected, of which 492 fall within the $150\mathrm{\mu m}\times 150\mathrm{\mu m}$ region shown, over which the image is centered.

Fig. 2. Poisson data model example. (a) The mean pixelated image of the in-focus and stationary molecule of Fig. 1, computed over an $11\times 11$ -pixel region with a pixel size of $13\mathrm{\mu m}\times 13\mathrm{\mu m}$ and a mean background photoelectron count of ${\beta }_{\theta ,0}=10$ per pixel (obtained as the product of the constant background photon detection rate ${\mathrm{\Lambda }}_0^b=24,200\text{photons}/\mathrm{s}$ and the uniform spatial distribution $f_{\theta }^b(x,y)=1/20,449{\mathrm{\mu m}}^{-2}$ , integrated over each pixel and the acquisition time of $t-t_0=0.05\mathrm{s}$ ). The molecule is located at $(x_0,y_0)=(656.5\mathrm{nm},682.5\mathrm{nm})$ in the object space, which corresponds to (5.05 pixels, 5.25 pixels) in the image space, assuming (0, 0) in both spaces coincides with the upper-left corner of the $11\times 11$ -pixel region. In terms of the photoelectron count due to the molecule, the mean image contains 476.15 out of the total of ${\mathrm{\Lambda }}_0·(t-t_0)=500$ mean number of photoelectrons distributed over the detector plane. All other relevant parameters are as specified in Fig. 1. (b) A Poisson realization of the mean image in (a). The photoelectron count in each pixel is drawn from the Poisson distribution with mean given by the photoelectron count in the corresponding pixel in (a).

Fig. 3. CCD data model and EMCCD data model examples. (a) A CCD realization of the mean image in Fig. 2(a). The electron count in each pixel is the sum of a random number drawn from the Poisson distribution with mean given by the photoelectron count in the corresponding pixel in Fig. 2(a), and a random number drawn from the Gaussian distribution with mean ${\eta }_0=0$ electrons and standard deviation ${\sigma }_0=8$ electrons. In this particular realization, the Poisson random numbers are taken directly from the Poisson realization of Fig. 2(b). (b) An EMCCD realization of the mean image in Fig. 2(a). The electron count in each pixel is the sum of a random number drawn from the probability distribution of Eq. (14) with electron multiplication gain $g=950$ and ${\nu }_{\theta ,k}$ given by the photoelectron count in the corresponding pixel in Fig. 2(a), and a random number drawn from the Gaussian distribution with mean ${\eta }_0=0$ electrons and standard deviation ${\sigma }_0=24$ electrons.

Fig. 4. Limits of accuracy under the fundamental data model. (a) Limit of accuracy ${\delta }_{x_0}$ for the estimation of the $x_0$ coordinate of an in-focus and stationary molecule from a fundamental image is plotted as a function of the expected number of photons detected in the image. Curves are shown that correspond to the detection of photons of wavelengths $\lambda =620\mathrm{nm}$ (dotted), 520 nm (solid), and 440 nm (dashed), which are collected by an objective lens with numerical aperture $n_a=1.4$ . The expected photon counts range from 100 to 10,000 and are obtained as the product of the constant photon detection rate ${\mathrm{\Lambda }}_0=10,000\text{photons}/\mathrm{s}$ and acquisition times $t-t_0$ ranging from 0.01 to 1 s. The image of the molecule is described by the Airy image function. (b) Limit of accuracy ${\delta }_{{\mathrm{\Lambda }}_0}$ for the estimation of the photon detection rate ${\mathrm{\Lambda }}_0=10,000\text{photons}/\mathrm{s}$ from a fundamental image is shown as a function of the expected number of photons detected in the image.

Fig. 5. Limit of the accuracy for separation distance estimation under the fundamental data model. The limit of accuracy ${\delta }_d$ for the estimation, from a fundamental image, of the distance $d$ separating two in-focus and stationary molecules is plotted as a function of the distance $d$ . Curves are shown that correspond to the detection of photons of wavelengths $\lambda =620\mathrm{nm}$ (⋄), 520 nm (*), and 440 nm (○), which are collected by an objective lens with numerical aperture $n_a=1.4$ . Photons are detected from each molecule at a rate of ${\mathrm{\Lambda }}_0=10,000\text{photons}/\mathrm{s}$ over an acquisition time of 0.3 s, such that ${\mathrm{\Lambda }}_0·(t-t_0)=3000$ photons are, on average, detected from each molecule in a given image. The image of each molecule is described by the Airy image function. Inset shows the portion of the curves from $d=1\mathrm{nm}$ to $d=50\mathrm{nm}$ .

Fig. 6. Comparing limits of accuracy corresponding to different data models and variations thereof. (a) Limit of accuracy ${\delta }_{x_0}$ for the estimation of the $x_0$ coordinate of an in-focus and stationary molecule from an image is plotted as a function of the effective pixel size for the Poisson data model (*) and the Poisson data model assuming the absence of background noise (+). Also shown are the limits of accuracy corresponding to the finite detection area variation of the fundamental data model (dashed line), the finite detection area variation of the fundamental data model assuming the absence of background noise (dotted line), and the fundamental data model (solid line), which are plotted as horizontal lines, as they do not depend on the effective pixel size. For the two cases involving the Poisson data model, the physical pixel size is $13\mathrm{\mu m}\times 13\mathrm{\mu m}$ , and the different effective pixel sizes are obtained by varying the lateral magnification from $M=20$ to $M=966.67$ . The image consists of a $15\times 15$ -pixel region at $M=100$ and a proportionately scaled pixel region at each of the other magnifications. At each magnification, the molecule is positioned such that the center of its image, given by the Airy image function, is located at 0.05 pixels in the $x$ direction and 0.25 pixels in the $y$ direction with respect to the upper-left corner of the center pixel of the image. For the two cases involving the finite detection area variation of the fundamental data model, ${\delta }_{x_0}$ is computed using the unpixelated equivalent of the $M=100$ scenario, though any other scenario can be used to obtain the same result. For all cases, photons are detected from the molecule at a rate of ${\mathrm{\Lambda }}_0=6000\text{photons}/\mathrm{s}$ over an acquisition time of $t-t_0=0.05\mathrm{s}$ , such that ${\mathrm{\Lambda }}_0·(t-t_0)=300$ photons are, on average, detected from the molecule over the detector plane (and 289.5 photons are, on average, detected from the molecule over a given finite-sized image). For the two cases involving a background component, a background detection rate of ${\mathrm{\Lambda }}_0^b=22,500\text{photons}/\mathrm{s}$ is assumed, along with a uniform background spatial distribution of $f^b(x,y)=1/A{\mathrm{\mu m}}^{-2}$ , $(x,y)\in C$ , where $C$ is the region in the detector plane corresponding to the image and $A$ is the area of $C$ . For example, for the $M=100$ scenario, $f^b(x,y)=1/38,025{\mathrm{\mu m}}^{-2}$ , such that there is a background level of ${\beta }_0=5$ photoelectrons per pixel over the $15\times 15$ -pixel region. For all cases, it is assumed that photons of wavelength $\lambda =520\mathrm{nm}$ are collected by an objective lens with numerical aperture $n_a=1.4$ .

Fig. 7. Comparing limits of accuracy corresponding to the CCD and EMCCD data models. (a) For the CCD data model, the limit of accuracy ${\delta }_{x_0}$ for the estimation of the $x_0$ coordinate of an in-focus and stationary molecule from an image is plotted as a function of the effective pixel size. Limits of accuracy are shown for three different levels of readout noise in each pixel, characterized by standard deviations of ${\sigma }_0=6$ electrons (•), 1 electron (⋄), and 0.5 electrons (○). In each case, the mean of the readout noise in each pixel is ${\eta }_0=0$ electrons. The limit of accuracy for the Poisson data model (with background noise) from Fig. 6 (*) is shown as a reference for comparison because the CCD limits of accuracy are computed by adding readout noise to the same assumed conditions. All details not mentioned are therefore exactly as given in Fig. 6 for the Poisson data model. The inset provides a clearer view of some data points for five relatively large effective pixel sizes. (b) Zoomed-in version of (a) with the addition of ${\delta }_{x_0}$ corresponding to the EMCCD data model with readout noise of mean ${\eta }_0=0$ electrons and standard deviation ${\sigma }_0=48$ electrons in each pixel and an electron multiplication gain of $g=1000$ .

Fig. 8. Comparing noise coefficients corresponding to the CCD and EMCCD data models. The noise coefficient ${\alpha }_k$ for the $k$ th pixel of an image is plotted as a function of the mean photoelectron count ${\nu }_{\theta ,k}$ in the pixel. Noise coefficients are shown for two EMCCD detectors operated at an electron multiplication gain of $g=1000$ , with readout noise of standard deviations ${\sigma }_0=24$ electrons (□) and 48 electrons (+) in each pixel, and for two CCD detectors, with readout noise of standard deviations ${\sigma }_0=0.5$ electrons (○) and 1 electron (⋄) in each pixel. The readout noise mean is ${\eta }_0=0$ electrons in all cases.

Fig. 9. Limits of the axial coordinate estimation accuracy for a conventional microscopy setup and a two-plane MUM setup. The limit of accuracy ${\delta }_{z_0}$ for the estimation of the $z_0$ coordinate of an out-of-focus and stationary molecule from a conventional image (*) or a pair of two-plane MUM images (○) is plotted as a function of the coordinate $z_0$ . Focal planes 1 and 2 of the MUM setup are denoted by vertical dashed lines and are located at $z_0=0\mathrm{nm}$ and $z_0=450\mathrm{nm}$ , respectively. The value of $z_0$ is specified with respect to focal plane 1, which coincides with the focal plane of the conventional setup. In both setups, an image consists of an $11\times 11$ -pixel region, acquired using a CCD detector with a pixel size of $13\mathrm{\mu m}\times 13\mathrm{\mu m}$ and readout noise with mean ${\eta }_0=0$ electrons and standard deviation ${\sigma }_0=6$ electrons at each pixel. The image of the molecule is described by the Born and Wolf image function. For the conventional setup, the molecule is laterally positioned such that the center of its image is located at 5.05 pixels in the $x$ direction and 5.25 pixels in the $y$ direction. Photons are detected from the molecule at a rate of ${\mathrm{\Lambda }}_0=10,000\text{photons}/\mathrm{s}$ over an acquisition time of $t-t_0=0.1\mathrm{s}$ , such that ${\mathrm{\Lambda }}_0·(t-t_0)=1000$ photons are on average detected from the molecule over the detector plane (and 952.3 photons are on average detected from the molecule over a given image when the molecule is in focus). A background detection rate of ${\mathrm{\Lambda }}_0^b=24,200\text{photons}/\mathrm{s}$ is assumed, along with a uniform background spatial distribution of $f^b(x,y)=1/20,449{\mathrm{\mu m}}^{-2}$ , such that there is a background level of ${\beta }_0=20$ photoelectrons per pixel over the $11\times 11$ -pixel image. Photons of wavelength $\lambda =520\mathrm{nm}$ are collected by an objective lens with magnification $M=100$ and numerical aperture $n_a=1.4$ . The refractive index of the immersion medium is $n=1.518$ . For the MUM setup, the collected molecule and background photons are split equally between the images corresponding to the two focal planes. Therefore, the photon detection rate is ${\mathrm{\Lambda }}_0=5000\text{photons}/\mathrm{s}$ per image, and the background detection rate is ${\mathrm{\Lambda }}_0^b=12,100\text{photons}/\mathrm{s}$ per image. For the image corresponding to focal plane 1, all other details are as given for the conventional setup. For the image corresponding to focal plane 2, the magnification is $M=98.18$ , calculated using a formula from [76] for a plane spacing of $\mathrm{\Delta }{z}_f=450\mathrm{nm}$ and a tube length of 160 mm. The lateral position of the image of the molecule and the background spatial distribution are accordingly adjusted based on this slightly smaller magnification.