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### Fluorescence microscopy: A statistics-optics perspective

##### Mohamadreza Fazel, Kristin S. Grussmayer, Boris Ferdman, Aleksandra Radenovic, Yoav Shechtman, Jörg Enderlein, and Steve Pressé

##### Rev. Mod. Phys. **96**, 025003 – Published 5 June 2024

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#### Abstract

Fundamental properties of light unavoidably impose features on images collected using fluorescence microscopes. Accounting for these features is often critical in quantitatively interpreting microscopy images, especially those gathering information at scales on par with or smaller than light’s emission wavelength. Here the optics responsible for generating fluorescent images, fluorophore properties, and microscopy modalities leveraging properties of both light and fluorophores, in addition to the necessarily probabilistic modeling tools imposed by the stochastic nature of light and measurement, are reviewed.

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- Received 14 February 2023

DOI:https://doi.org/10.1103/RevModPhys.96.025003

© 2024 American Physical Society

#### Physics Subject Headings (PhySH)

- Research Areas

Fluorescent biomolecules

- Techniques

Markovian processesOptical microscopyOptical techniquesSingle molecule techniquesStatistical methods

Physics of Living Systems

#### Authors & Affiliations

Mohamadreza Fazel

^{}Department of Physics, Arizona State University, Tempe, Arizona, USA and Center for Biological Physics, Arizona State University, Tempe, Arizona, USA

Kristin S. Grussmayer

^{}Department of Bionanoscience, Faculty of Applied Science and Kavli Institute for Nanoscience, Delft University of Technology, Delft, Netherlands

Boris Ferdman

^{}Russel Berrie Nanotechnology Institute and Department of Biomedical Engineering, Technion—Israel Institute of Technology, Haifa, Israel

Aleksandra Radenovic

^{}Laboratory of Nanoscale Biology, Institute of Bioengineering, Ecole Polytechnique Federale de Lausanne (EPFL), Lausanne, Switzerland

Yoav Shechtman

^{}Russel Berrie Nanotechnology Institute and Department of Biomedical Engineering, Technion—Israel Institute of Technology, Haifa, Israel

^{}III. Institute of Physics—Biophysics, Georg August University, Göttingen, Germany

^{}Department of Physics, Arizona State University, Tempe, Arizona, USA, Center for Biological Physics, Arizona State University, Tempe, Arizona, USA, and School of Molecular Sciences, Arizona State University, Tempe, Arizona, USA

^{*}Corresponding author: spresse@asu.edu

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Vol. 96, Iss. 2 — April - June 2024

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#### Images

###### Figure 1

Schematic of an infinity-corrected wide-field microscope consisting of an ideal objective lens with focal length ${f}_{1}$ and an ideal tube lens with focal length ${f}_{2}$. We show light propagation from a point source in the focal plane (sample space) to the image point in image space. The plane between the lenses, a distance ${f}_{1}$ away from the objective lens and a distance ${f}_{2}$ from the tube lens, is called the conjugate plane (the green vertical line). The conjugate plane is also sometimes termed the back focal plane, Fourier plane, or pupil plane. Here the light from any point source on the focal plane crosses through the same lateral position. By considerations of geometric proportion, the ratio of the lateral displacement of the image point to the lateral displacement of the source point is equal to the ratio of the focal lengths ${f}_{2}/{f}_{1}$. This ratio is the microscope’s magnification $\mathcal{M}$.

###### Figure 2

Visualization of the diffraction limit of resolution. Displayed are the interference patterns of two coherently emitting point emitters, shown by red dots, for three different distances between emitters across panels. The closer the emitters are positioned with respect to each other, the larger the angular positions of the destructive interference lanes (directions of zero light intensity). At a critical distance, shown in the right panel, the first lane of destructive interference is positioned at the half angle $\mathrm{\Theta}$ of light collection of the objective, and the objective lens receives a continuous wave front absent intensity minima appearing as a single emitter wave front.

###### Figure 3

Lateral resolution limit of a CLSM. The resolution is determined by the highest lateral spatial frequency contained in a focused bright spot. This is generated by the interference of two rays traveling from the edges of the objective to the focal point with the highest possible incidence angle $\mathrm{\Theta}$ with respect to the optical axis as shown. The associated wave vectors are of equal magnitude $2\pi n/\lambda $, where $\lambda $ is the vacuum wavelength. The corresponding lateral components ${k}_{x,\theta}$ of these wave vectors are of equal magnitude given by ${k}_{x,\theta}=2\pi n\mathrm{sin}\mathrm{\Theta}/\lambda $ and opposite directions resulting in a difference of $4\pi n\mathrm{sin}\mathrm{\Theta}/\lambda $. As such, the interference of the two beams leads to a periodic interference pattern in the lateral direction with a periodicity $\lambda /2n\mathrm{sin}\mathrm{\Theta}$ that is equal to the lateral resolution limit of a CLSM.

###### Figure 4

Axial resolution of a CLSM. Like the lateral resolution, the axial resolution is determined by the tightest spatial modulation of light that can be generated along the optical axis. This is achieved by interfering an axially propagating beam with one traveling at the highest possible incidence angle. The axial component of the wave vector of the former is equal to the full wave-vector length ${k}_{0}=2\pi n/\lambda $, and the axial component for the latter is ${k}_{z,\mathrm{\Theta}}=2\pi n\mathrm{cos}\mathrm{\Theta}/\lambda $. The resulting interference therefore leads to a spatial intensity modulation along the optical axis witha periodicity $\lambda /n(1-\mathrm{cos}\mathrm{\Theta})$ setting a CLSM’s axial resolution limit.

###### Figure 5

Lateral and axial resolution in diffraction-limited optical microscopy using a water immersion objective (designed for imaging in water with a refractive index of 1.33) as a function of numerical aperture (NA) and wavelength.

###### Figure 6

Simplified Jablonski diagram. The electronic ground state ${S}_{0}$, the singlet excited states ${S}_{n}$, the triplet excited states ${T}_{n}$, and radical cation ${F}^{\xb7+}$ or anion states ${F}^{\xb7-}$. The thick lines represent electronic energy levels, the thin lines indicate vibrational energy levels, and rotational energy states are left unmarked. $P$, phosphorescence; VR, vibrational relaxation; IC, internal conversion; ISC, intersystem crossing. The rates of oxidation and reduction are ${k}_{\mathrm{ox}}$ and ${k}_{\mathrm{red}}$, respectively. The arrows represent a subsample of all possible transitions between different states.

###### Figure 8

The optical microscope, i.e., the imaging system, is a wave front transforming system converting (left sketch)the outgoing spherical wave front of a point emitter in sample space into (right sketch)a concentric spherical wave front in image space converging into an image point in the image space.

###### Figure 9

The phase relation between planar wave front segments propagating along the same angle $\theta $ but emanating from two different point sources, where one point source is on the optical axis (red) and the other is laterally shifted by a distance $y$ (green). The image point (the point of convergence of the spherical wave front segment) corresponding to the shifted point source is translated by a distance ${y}^{\prime}$ away from the optical axis. The ratio between ${y}^{\prime}$ and $y$ is the magnification $\mathcal{M}$. Optical path length differences between wave front segments traveling along angles $\theta $ and ${\theta}^{\prime}$, respectively, are shown as thin blue lines at the emitters’ positions and oriented perpendicularly to the propagation directions $\theta $ and ${\theta}^{\prime}$.

###### Figure 10

Phase relation between planar wave front segments propagating along the same angle $\theta $ but emanating from two different point sources along the optical axis. As in Fig.9, optical path differences (phase differences) between wave front segments traveling along angles $\theta $ and ${\theta}^{\prime}$, respectively, are shown as blue rectangles.

###### Figure 11

Geometry of propagation of a narrow section of the wave front from the emitter to the image plane.

###### Figure 12

From electric or magnetic field to intensity. Left panel:two spherical caps show the support of the Fourier representations of electric and magnetic fields given by Eq.(44). Right panel:representation of the extent of frequency support of the imaging OTF obtained by the convolution of the two caps in the left panel; see Eq.(50). The shape of the right panel is termed the butterfly shape, and its missing cone in the middle highlights a wide-field microscope’s inability to collect sufficient axial frequencies and thus a lack of optical sectioning.

###### Figure 13

Visualization of the maximum axial and lateral extents of the Fourier representation of the electric field and the imaging OTF. (a)Cross section of a Fourier representation of the electric field (cap) at ${k}_{y}^{\prime}=0$. The cross section is an arc with radius ${k}^{\prime}=2\pi /\lambda $ and $0\le {\theta}^{\prime}<{\mathrm{\Theta}}^{\prime}$; see Eq.(45). The maximum extents of the cap along the lateral and axial directions are given by $\mathrm{\Delta}{k}_{\parallel}^{\prime}=(2\pi /\lambda )\mathrm{sin}{\mathrm{\Theta}}^{\prime}$ and $\mathrm{\Delta}{k}_{z}^{\prime}=(2\pi /\lambda )(1-\mathrm{cos}{\mathrm{\Theta}}^{\prime})$, respectively. (b)Convolution of the caps associated with the electric and magnetic fields along the largest axial and lateral extents beyond which the convolution is zero.

###### Figure 14

Angular distribution of the electric field generated by a single dipole emitter. The gray rectangle represents the cover slide (commonly assumed to coincide with the $z=0$ plane), which is the interface between the electric dipole’s embedding medium (above the cover slide) and the immersion medium below the cover slide. The red two-headed arrow depicts the dipole; $\alpha $ and $\beta $ are the polar and inclination (azimuthal) angles describing the orientation of the dipole, respectively, $\varphi $ is the polar angle of the wave vector, and ${\theta}_{d}$ and $\theta $ are the azimuthal angles of the wave vector above and below the interface.

###### Figure 15

PSF of a wide-field microscope projected into sample space. Shown are plots of the $1/e$, $1/{e}^{2}$, and $1/{e}^{3}$ isosurfaces of the maximum PSF value. The lateral coordinates refer to back-projected sample space coordinates $(x,y)=({x}^{\prime},{y}^{\prime})/\mathcal{M}$, whereas the axial coordinate refers to an emitter’s axial position ${z}_{d}$. We retain this PSF representation throughout the review. The panels are described in the text. Calculations were performed for a $\mathrm{NA}=1.2$ water immersion objective with $n=1.33$ and an emission wavelength $\lambda =550\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$.

###### Figure 16

Effect of orientation on the emitter’s image. Top row:images of electric dipole emitters of fixed strength but different orientations in the $x\text{\u2212}z$ plane, where $\beta $ is the inclination angle; see Fig.14. The emitter is situated 400nm below the focal plane ($\mathrm{NA}=1.2$, $n=1.33$). Middle row:same as the top row, but for an emitter situated in the focal plane. Bottom row:same again but for an emitter situated 400nm above the focal plane. The scale bar is $0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}\mathrm{m}$.

###### Figure 17

Effect of a refractive index mismatch on the PSF. The PSF of a rapidly rotating electric dipole emitter (isotropic emitter) is positioned at various distances from a cover slide surface ($z=0$). The calculations were done for an $\mathrm{NA}=1.2$ objective corrected for an immersion or medium with $n=1.33$, while the solution above the cover slide has $n=1.38$ (i.e., a refractive index mismatch $\mathrm{\Delta}n=0.05$). The bottom of each box shows a density plot of the PSF’s cross section through its maximum value.

###### Figure 18

Comparison between scalar and vector PSF calculations. Shown are cross sections of the PSF across the $x$ axis in the focal plane. The red curve shows the results of the full wave-vector PSF calculation for an electric dipole emitter with a fixed $x$-axis orientation, the blue curve displays the same calculation for a rapidly rotating isotropic or random emitter, the green curve presents the result of Eq.(66), and the ochre curve shows the Gaussian approximation of Eq.(69). Left inset:three-dimensional isosurface PSF plot using the exact vector field calculation for an isotropic emitter. Right inset:three-dimensional isosurface PSF plot using the exact vector field calculation for the scalar approximation. All calculations were performed for a water immersion objective with $\mathrm{NA}=1.2$.

###### Figure 19

Scalar approximation of the OTF of a wide-field microscope. Calculations were done for an $\mathrm{NA}=1.2$ water immersion objective and an emission wavelength of 550nm. Left panel:${k}_{x}{k}_{z}$ cross section of the electric field amplitudes in

*sample space*having a frequency support (frequencies with nonzero amplitude) in the shape of a spherical cap with radius $k=2\pi n/\lambda $ and an opening half angle equal to the objective’s maximum half angle $\mathrm{\Theta}$. Middle panel:the same distribution for the magnetic field. Right panel:three-dimensional convolution of the left two panels yielding the scalar approximation of the OTF amplitude. All panels show density plots of the*decadic logarithm*of the Fourier amplitude’s absolute value (see the color bar on the right-hand side) normalized by the maximum absolute value of the corresponding amplitudes. In all panels, the coordinate origin (${k}_{x}=0$ and ${k}_{z}=0$) is at the center. Throughout this review, we use the same representation for all OTFs shown.###### Figure 20

Density plots for the first 12 Zernike polynomials presented in Table1: (1)horizontal or $x$ tilt, (2)vertical or $y$ tilt, (3)defocus, (4)vertical astigmatism, (5)oblique astigmatism, (6)horizontal coma, (7)vertical coma, (8)primary spherical aberration, (9)oblique trefoil, (10)vertical trefoil, (11)vertical secondary astigmatism, and (12)oblique secondary astigmatism.

###### Figure 21

Model calculations of the image of an isotropic emitter (a rapidly rotating dipole emitter) aberrated by a phase function given by the Zernike polynomials shown in Fig.20. To better visualize the effects of aberration, all Zernike polynomials were multiplied by a factor of 2.5. Calculations were again done for a water immersion objective with $\mathrm{NA}=1.2$ and for an emission wavelength of 550nm. The yellow scale bar is $0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}\mathrm{m}$.

###### Figure 22

TIRF microscopy. The excitation intensity above a cover slide interface with the sample medium is displayed as a function of incidence angle. The sample solution and cover slide refractive indices are 1.33 (water) and 1.52, respectively, resulting in a TIR critical angle of $\approx 61\xb0$. The excitation wavelength is taken as 470nm.

###### Figure 23

Supercritical angle fluorescence (SAF) microscopy. Ratio of supercritical to total downward fluorescence emission for a rapidly rotating molecule as a function of distance from the interface of the cover slide and the sample medium. The refractive indices of the sample solution and cover slide are assumed to be 1.33 (water) and 1.52 (glass), respectively, with an emission wavelength of 550nm. Inset:angular emission intensity distribution of an emitter directly on the interface (with the blue, red, and green curves denoting the UAF and SAF emissions and the emission toward the sample solution, respectively). The SAF emission strongly depends on the emitter’s distance to the interface, while the undercritical emission is independent of the emitter axial position. By determining the ratio of SAF to SAF plus UAF emission, we can find the axial position of an emitter.

###### Figure 24

MIET microscopy: Dependence of the fluorescence lifetime (in terms of the free space lifetime ${\tau}_{0}$) on the emitter’s distance from the glass substrate (cover slide) coated with a 20nm gold layer. Calculations were done for an emission wavelength of 550nm and a unit fluorescence quantum yield. Free curves for the vertical, horizontal, and random emission dipole orientations are shown. Inset:MIET sample geometry.

###### Figure 25

Geometry for deriving the electric field generated by a single dipole emitter above the MIET substrate (the metal surface). The red double-headed arrow shows a dipole located a distance ${z}_{d}$ above the metal surface with an orientation of $\beta $ and $\alpha $ denoting the polar and inclination azimuthal angles, respectively. The three longer single-headed arrows show plane wave component vectors with the corresponding perpendicular polarization unit vectors ${\widehat{\mathbf{e}}}_{\parallel}$ and ${\widehat{\mathbf{e}}}_{\perp}^{\pm}$. Here ${\widehat{\mathbf{e}}}_{\perp}^{+}$ is the unit vector associated with the wave vector moving toward the metal surface. Similar conventions hold for the other unit vectors.

###### Figure 26

Schematic of a CLSM. The yellow and red beams show the excitation and emission light, respectively. The emission passes through a confocal pinhole suppressing out-of-focus light; see the text for details.

###### Figure 27

Schematic of the geometry when a planar laser wave front is focused through the objective into the sample space; see Fig.26. Wave front patches at distance $\rho $ from the optical axis in the back focal plane are converted into spherical wave front patches traveling at an angle $\theta =\mathrm{arcsin}(\rho /nf)$ with respect to the optical axis $z$, where $f$ is the focal length of the objective lens; see the text for details.

###### Figure 28

CLSM and STED intensity distributions at the focus. Shown is a comparison of intensity distribution (left panels)with a conventional CLSM focus, (middle panels)with a $z$-STED focus, and (right panels)with an $xy$-STED focus. Calculations were done for a water immersion objective with $\mathrm{NA}=1.2$ at an excitation wavelength of 470nm. The excitation polarization and its generating phase plate are shown in the top panels. The bottom panels show 3D contour plots of the $1/e$, $1/{e}^{2}$, and $1/{e}^{3}$ intensity isosurfaces and projections of $x\text{\u2212}y$, $x\text{\u2212}z$, and $y\text{\u2212}z$ cross sections through the center.

###### Figure 29

Anatomy of the OTF (amplitude) of a confocal microscope. Left panel:excitation OTF. Middle panel:detection OTF for a confocal pinhole with a $50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}\mathrm{m}$ radius and 60 times magnification. Right panel:the resulting confocal OTF obtained via a 3D convolution of the two leftmost distributions.

###### Figure 30

OTF amplitude of a confocal microscope as a function of confocal aperture size. The confocal aperture radius is given in the top row. We assumed an excitation wavelength of 470nm, an emission wavelength of 550nm, and a water immersion objective of $\mathrm{NA}=1.2$ at 60 times magnification. The top left panel shows the limit of an extremely large confocal pinhole in which the OTF approaches that of a wide-field microscope imaging at the same wavelength as the excitation wavelength of the excitation laser. The bottom right panel shows the limit of a nearly zero-size pinhole ($a=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}\mathrm{m}$) in which the OTF approaches that of an ISM; see Sec.4b2.

###### Figure 31

Confocal microscope PSF for an isotropic emitter as a function of confocal aperture size. The aperture radius is given above each panel. The parameters are similar to those in Fig.30, with 60 times magnification.

###### Figure 32

Relationship between PSF size and detection efficiency in a CLSM. We show the light detection efficiency vs the Gaussian radius $\sigma $ of the PSF in the focal plane as a function of the confocal aperture’s radius (denoted as $a$). Calculations were made for a water immersion objective with $\mathrm{NA}=1.2$ and an image magnification of 60 times (focal plane to pinhole plane). It was assumed that excitation is achieved with 470nm circular polarized light focused onto a diffraction-limited spot, and that the fluorescence emission is of 550nm wavelength. We found the focal radius by fitting a radially symmetric Gaussian approximation $\mathrm{exp}(-{\rho}^{2}/2{\sigma}^{2})$ to the PSF in the focal plane. The curve’s undulations at the upper right arise from the diffraction effects of light passing through a circular pinhole.

###### Figure 33

Image formation in ISM. The blue curve represents the excitation intensity distribution ${I}_{\mathrm{ex}}$ (excitation PSF) with its center at $\mathit{\xi}=0$ (the optical axis). The yellow curve shows the detection PSF (${U}_{\mathrm{wf}}$) for a pixel located at $\mathit{\xi}$ away from the optical axis. The pixel PSF (${U}_{\mathrm{pix}}$) describing the image formation is, however, given by the product of the excitation and detection PSFs, which is designated by the green curve and centered at $\mathit{\xi}/\kappa $. Thus, a fluorophore at $\mathit{\xi}=0$ (the excitation intensity’s center) will appear at $\mathit{\xi}/\kappa $.

###### Figure 34

ISM image reconstruction. Top image:at each scan position, the array detector records a small image of the illuminated region. To reconstruct a final ISM image, we can either downscale each recorded small image by a factor of $\kappa $ (bottom right image) or leave the recorded images unchanged but place them in the final ISM image by a factor of $\kappa $ farther way from each other (bottom left image).

###### Figure 35

4Pi microscope excitation OTF generated by the interference of light focused through two opposing objectives. The left and middle panels show the same Fourier transform of the excitation electric field in sample space. The resulting excitation OTF shown in the right panel is the (auto)convolution of this electric field Fourier transform and represents the Fourier transform of the excitation intensity (the excitation OTF). Excitation is assumed to be done using a water immersion objective with $\mathrm{NA}=1.2$.

###### Figure 36

Excitation PSF and imaging PSF of 4Pi microscopy for a rapidly rotating emitter. Left panel:excitation PSF in the focus of a 4Pi microscope. Middle panel:imaging PSF of a 4Pi type A microscope. Right panel:imagine PSF of a 4Pi type C microscope. Calculations were performed using a water immersion objective with an $\mathrm{NA}=1.2$, a 470nm excitation wavelength, and a 550nm fluorescence emission wavelength, and for a confocal detection in the limit of an infinitely small pinhole.

###### Figure 37

OTF of a type A 4Pi microscope where excitation is done through two opposing objectives, and detection is performed from one side through a confocal pinhole. For simplicity we consider here only the limiting case of an infinitely small pinhole maximizing the spatial resolution. Left panel:excitation OTF. Middle panel:the OTF of detection with an infinitely small pinhole. Right panel:the resulting 4Pi OTF as a convolution of the two distributions shown on the left. Excitation and detection are achieved using a water immersion objective with an $\mathrm{NA}=1.2$, and any Stokes shift between excitation and emission light is neglected.

###### Figure 38

OTF of a type C 4Pi microscope. Like Fig.37, but in this configuration both excitation and detection occur through two opposing objectives. Again we consider here only the limiting case of an infinitely small pinhole. Left panel:excitation OTF. Middle panel:the identical Fourier transform for coherent confocal detection from both sides. Right panel:the resulting OTF as a convolution of the two panels shown on the left.

###### Figure 39

Pixel reassignment in two-photon-excitation ISM. In contrast to the ISM in Fig.33, the excitation intensity distribution (one-photon-excitation PSF) in two-photon microscopy has a larger width due to its larger excitation wavelength.

###### Figure 41

(a)Schematic of confocal volume (in blue) with labeled molecules emitting photons in proportion to their degree of excitation decaying from the confocal volume center. (b)Synthetic trace with 1500 photons generated assuming four molecules diffusing at $1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}{\mathrm{m}}^{2}/\mathrm{s}$ for 30ms using background and molecule photon emission rates of ${10}^{3}$ and $4\times {10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{photons}/\mathrm{s}$, respectively. Adapted from [461].

###### Figure 43

Comparison of the diffusion coefficients $D$ obtained from the statistical framework vs FCS plotted against photon counts used in the analysis. The photon arrival times were simulated using the parameter values in Fig.41. Adapted from [461].

###### Figure 44

Multifocal setup uniquely resolving many molecular trajectories simultaneously. (a)A beam splitter is used to divide the fluorescent emission (indicated in green) into two paths later coupled into fibers and detected by four APDs corresponding to different focal spots. (b)PSFs associated with different light paths. (c)Trajectories for two freely diffusing molecules with $D=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}{\mathrm{m}}^{2}/\mathrm{s}$, ${\mu}_{0}=5\times {10}^{4}\text{photons/s}$, and ${\mu}_{\mathcal{B}}={10}^{3}\text{photons/s}$. The orange and blue curves represent the learned trajectories’ ground truth and median, respectively. The blue and gray areas, respectively, denote the 95% confidence intervals and the PSF’s width. Adapted from [235].

###### Figure 45

Lifetime histograms from single-pixel FLIM. Lifetimes are below the IRF and differ by subnanoseconds. (a)–(c)Datasets simulated with $5\times {10}^{2}$, ${10}^{3}$, and $2\times {10}^{3}\text{photons}$, an IRF width of 0.66ns, and ground truth lifetimes of 0.2 and 0.6ns denoted by dotted lines. Learning the correct number of fluorophore species here requires $>500\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{photons}$.

###### Figure 47

Experimental FLIM data from mixtures of two cellular structures (lysosome and mitochondria, shown in green and red, respectively) stained with two different fluorophore species. (a),(b)Ground truth lifetime maps. (c)Data acquired from mixtures of two ground truth maps. (d),(e)Resulting subpixel interpolated lifetime maps obtained using the statistical framework of Fig.48. The average absolute difference between ground the truth and learned maps is $\approx 4\%$. Scale bars are $4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}\mathrm{m}$. Adapted from [134].

###### Figure 49

Sinusoidal illumination pattern for SIM microscopy. ${\mathbf{k}}_{i}$ is the wave vector, $L$ is the fringe spacing, and ${\gamma}_{i}$ is the illumination’s in-plane angle. The phase is related to the position of the maxima relative to the optical axis.

###### Figure 50

SIM OTF. Left panel:Fourier transform of the modulated illumination intensity (SIM excitation OTF given by the three delta peaks). Middle panel:Fourier transform of the wide-field detection. Right panel:SIM OTF obtained by convolution of the other two panels; see also Eq.(118).

###### Figure 52

LSFM setups. (a)In digitally scanned laser light-sheet microscopy (DLSM) a galvanometric (galvo) scanning unit rapidly moves a Gaussian beam perpendicular to the detection axis focused in the sample through the excitation objective lens (${\mathrm{OL}}_{\mathrm{ex}}$). Signal from the focal plane is collected through the detection objective lens (${\mathrm{OL}}_{\mathrm{det}}$) and tube lens (TL) onto a camera ($C$). (b)In SPIM, a static light sheet is formed when a cylindrical lens in the excitation path creates an elongated beam in one direction and the same perpendicular detection optics as in (a). (c)Schematic of the Gaussian beam in (a) and (b) focused through a lens or objective with a diameter $D$, a beam waist ${\omega}_{0}$, and a Raleigh length ${z}_{r}$.

###### Figure 53

SPIM OTF. Excitation is achieved by focusing a plane wave through a low-aperture lens ($\mathrm{NA}=0.4$) from the left, resulting in a weakly diverging horizontally elongated excitation region. See the text for further details.

###### Figure 54

Multiplane microscopy. (a)A conventional fluorescence microscope with epifluorescence (FL) and white light illumination (IL) acquire images of different focal planes across the sample by moving the objective lens (OL) and the sample with respect to each other. The nominal focal plane is shown in black, while the planes shown in red and blue can be imaged by adjusting the axial positions of the sample. Shown are the sample ($S$), the objective lens (OL), the dichroic mirror (DM), and the tube lens (YL). (b)A multiplane microscope relays the optical path from the intermediate image formed in the panel via a telescope with lenses of focal lengths ${F}_{1}$ and ${F}_{2}$ and uses a beam-splitting prism, i.e., a refractive element, along the detection path to separate fluorescence emission into multiple channels (here four) with different focal planes projected next to each other on two cameras ($C1$ and $C2$); see [111]. (c)A multifocus microscope uses a multifocus grating (MFG), i.e., a diffractive element, a chromatic correction grating (CCG), and a chromatic correction prism (CCP) to achieve multiple focal planes on one camera; see the text for more details.

###### Figure 55

Schematics for STED imaging. Excitation and depletion beams are used to acquire a subdiffraction-limited image, formed after raster scanning the full sample. The resulting image can be understood as a convolution between the effective PSF formed from the excitation and depletion laser beams and the fluorescent molecule distribution in the sample. Left panel:schematics comparing diffraction-limited confocal images of microtubules with the coinciding STED image. Right panels:electronic transitions of excitation, and stimulated emission in STED (top panel), ground-state depletion GSD (middle panel), and RESOLFT (bottom panels).

###### Figure 56

MINFLUX’s working principle. MINFLUX employs a doughnut-shaped excitation beam (orange) with the doughnut translated to four locations (blue circles) at which fluorescence signals are measured and used to determine the fluorophore’s position. The red and black stars indicate the excited- and ground-state fluorophores, respectively; see the text for details.

###### Figure 57

Single emitters are stochastically activated to become fluorescent. The activated emitters can be precisely localized provided that they are spaced farther apart than the Nyquist limit; see Sec.1c. The process is repeated for tens of thousands of frames. In each frame, single emitters are identified and fitted to obtain their center of mass, allowing superresolved pointillistic image reconstruction (bottom right panel). Repetitive activation, localization, and deactivation temporally separate spatially unresolved structures in a reconstructed image, with the apparent resolution gain compared to the standard diffraction-limited image (bottom row).

###### Figure 58

Imaging with DNA PAINT. (a)Schematics illustrate DNA PAINT where dye-conjugated oligo (imager oligo) transiently hybridizes with a complementary (docking) oligo. (b)The binding time ${\tau}_{B}$ (or the dissociation rate $1/{\tau}_{B}$) depends on the imager strand length. (c)Increasing either imager strand concentration or docking site density decreases the dark times ${\tau}_{D}$ (interevent lifetime). Adapted from [412].

###### Figure 61

PSF engineering. (a)Frequently used engineered PSFs, simulated for an objective lens with an $\mathrm{NA}=1.49$ and a pixel size of 110nm. The top row is the wide-field PSF. Other rows present commonly used phase masks and their corresponding PSFs over a range of axial positions. (b)CRLB (see Sec.1b) of the 3D position (each axis individually) plotted as a function of the axial position assuming that the system is laterally shift invariant. The subscripts in the axis labels indicate the coordinate for which the CRLB was calculated.

###### Figure 64

Single-photon detector. Laser pulses and their centers are shown as blue spikes and dashed red lines with an interpulse window $T$, respectively. The fluorophore excitation, photon emission, and photon detection events take place at ${t}_{\mathrm{ext}}$, ${t}_{\mathrm{ems}}$, and ${t}_{\mathrm{det}}$, respectively (designated as dashed black lines). The fluorophore spends time $\mathrm{\Delta}{t}_{\mathrm{ext}}$ in the excited state and emits a photon after $n$ pulses. The reported photon arrival time $\mathrm{\Delta}{t}_{k}$ is measured with respect to the immediate previous pulse center. Moreover, ${\mathrm{\Delta}}_{1}$ and ${\mathrm{\Delta}}_{2}$ denote the difference of the excitation pulse center and the detector delay in reporting the photon arrival time.

###### Figure 65

Fluorophore enumeration. (a)Sketch of the enumeration problem where the ROI intensity varies as fluorophores switch between the dark, bright, and photobleached states. (b)–(d)Histogram of the sampled posterior over the number of fluorophores, i.e., the sum of sampled loads, for experimental data with 24, 49, and 98 fluorophores, respectively, using the statistical framework appearing in Fig.66. Adapted from [59].

###### Figure 67

Data simulated for discrete measurements of two state systems with (a)fast and (b)slow transitions. The system trajectories in the state space, measurements at different times intervals ($\delta T$), i.e., bins, and the state signal levels in the absence of noise are denoted as cyan, gray, and dotted lines, respectively. The measurements between the state signal levels coincide with time intervals where the system has switched to a different state at some point during those intervals. In the simulations, data acquisitions take place at every $\delta T=0.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{s}$, where the average time spent in each state is 0.8 and 0.066s for slow and fast kinetics, respectively. Adapted from [254].

###### Figure 68

Contour for the integration over ${k}_{z}^{\prime}$ of Eq.(56) in the complex ${k}_{z}^{\prime}$ plane. For positive values of $z-{z}_{d}$, the contour has to be closed at infinity over the positive $\mathrm{Im}({k}_{z}^{\prime})$ half-space, while for negative values of $z-{z}_{d}$ it is closed at infinity over the negative half-space. Along the real axis, the integrand has two poles at $\pm {w}_{d}=\pm \sqrt{{k}_{d}^{2}-{q}^{2}}$.

###### Figure 69

Posteriors over diffusion coefficients strongly depend on the prespecified $M$ when operating within a parametric Bayesian paradigm. The trace analyzed contains $\approx 1800$photons generated from four molecules diffusing at $D=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu}{\mathrm{m}}^{2}/\mathrm{s}$ for 30ms with a background and maximum molecule photon emission rate of ${10}^{3}$ and $4\times {10}^{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{photons}/\mathrm{s}$, respectively. To deduce $D$ within the parametric paradigm, we assumed a fixed number of molecules: (a)$M=1$, (b)$M=2$, (c)$M=3$, (d)$M=4$, and (e)$M=5$. The correct estimate in (d) (and the mismatches in all others) highlights why we must use the available photons to simultaneously learn the number of molecules and $D$. Adapted from [461].