Fluorescence Microscopy: a statistics-optics perspective

Abstract

Fundamental properties of light unavoidably impose features on images collected using fluorescence microscopes. Modeling these features is ever more important in quantitatively interpreting microscopy images collected at scales on par or smaller than light's wavelength. Here we review the optics responsible for generating fluorescent images, fluorophore properties, 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.

FIG. 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 $f_2$ from the tube lens, is called the conjugate plane (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, it can be seen that the ratio of lateral displacement of the image point to 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 $ℳ$.

FIG. 2:

Visualization of the diffraction limit of resolution. Here, we show 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 $\Theta$ of light collection of the objective, and the objective lens receives a continuous wavefront absent intensity minima appearing as a single emitter wavefront.

FIG. 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 $\Theta$ 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 $k_{x,\theta }=2\pi n \text{sin}\Theta /\lambda$, and opposite directions resulting in a difference of $4\pi n \text{sin}\Theta /\lambda$. As such, the interference of the two beams leads to a periodic interference pattern in the lateral direction with periodicity $\lambda /2n \text{sin}\Theta$, equal to the lateral resolution limit of a CLSM.

FIG. 4:

Axial resolution of a CLSM: Similar to 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 $k_{z,\Theta }=2\pi n \text{cos}\Theta /\lambda$. The resulting interference therefore leads to a spatial intensity modulation along the optical axis with periodicity $\lambda /n(1-\text{cos}\Theta )$ setting a CLSM’s axial resolution limit.

FIG. 5:

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

FIG. 6:

Simplified Jablonski diagram. The electronic ground state $S_0$, the singlet excited states $S_n$, the triplet excited states ${\text{T}}_n$, and radical cation ${\text{F}}^{\cdot +}$ or anion states ${\text{F}}^{·–}$. Thick lines represent electronic energy levels, thin lines vibrational energy levels, while rotational energy states are left unmarked. Here we denote: Phosphorescence by P; Vibrational Relaxation by VR; Internal Conversion by IC; Inter System Crossing by ISC; and rates of oxidation and reduction are $k_{\text{ox}}$ and $k_{\text{red}}$, respectively. Arrows represent a subsample from all possible transitions between different states.

FIG. 7:

Data simulated for discrete measurements of two state systems with fast and slow transitions depicted in panels a and b, respectively. The system trajectories in the state space, measurements at different times intervals $\left(\delta T\right)$, i.e., bins, and the state signal levels in the absence of noise are, respectively, denoted by cyan, gray, and dotted lines. The measurements between the state signals level 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 $\delta T=0.1\text{ s}$ where the average time spent in each state is, respectively, 0.8 s and 0.066 s for slow and fast kinetics. The figure is adapted from Ref. [148].

FIG. 8:

Fluorophore enumeration. (a) Cartoon representation of the enumeration problem where the ROI intensity varies as fluorophores switch between the dark, bright, and photo-bleached states. (b-d) Histogram of the sampled posterior over the number of fluorophores, i.e., sum of sampled loads, for experimental data with, respectively, 24, 49 and 98 fluorophores using the statistical framework appearing in Box II C. The figure is adapted from Ref. [76].

FIG. 9:

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

FIG. 10:

The phase relation between planar wavefront 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 (point of convergence of the spherical wavefront 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 $ℳ$. Optical path length differences between wavefront segments traveling along angles $\theta$ or ${\theta }^{\prime }$, respectively, are shown as thin bluish lines at the emitters’ positions and oriented perpendicular to the propagation directions $\theta$ and ${\theta }^{\prime }$.

FIG. 11:

Phase relation between planar wavefront segments propagating along the same angle $\theta$ but emanating from two different point sources along the optical axis. Similar to Fig. 10, optical path differences (phase differences) between wavefront segments traveling along angles $\theta$ or ${\theta }^{\prime }$, respectively, are shown as blue rectangles.

FIG. 12:

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

FIG. 13:

From electric/magnetic field to intensity. The two spherical caps in the left panel show the support of the Fourier representations of electric and magnetic fields given by Eq. 49. The right panel represents the extent of frequency support of the imaging OTF obtained by the convolution of the two caps on the left panel; see Eq. 55. The shape in the right panel is termed butterfly-shape and its missing cone in the middle highlights a wide-field microscope’s inability to collect sufficient axial frequencies and thus lack of optical sectioning.

FIG. 14:

Visualization of the maximum axial and lateral extents of the Fourier representation of the electric field and the imaging OTF. (a) A cross-section of the 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 $0\le {\theta }^{\prime }<{\Theta }^{\prime }$ (see Eq. 50). The maximum extents of the cap along the lateral and axial directions are, respectively, given by $\Delta k_{\parallel }^{\prime }=\frac{2\pi }{\lambda }\text{sin}{\Theta }^{\prime }$ and $\Delta k_z^{\prime }=\frac{2\pi }{\lambda }\left(1-\text{cos}{\Theta }^{\prime }\right)$. (b) Here we show the convolution of the caps associated to the electric and magnetic fields along the largest axial and lateral extents beyond which the convolution is zero.

FIG. 15:

Contour for the integration over $k_z^{\prime }$ of Eq. 61 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 $\text{Im}\left(k_z^{\prime }\right)$ half-space, while for negative values of $z-z_d$ it is over the negative half-space. Along the real axis, the integrand has two poles at $\pm w_d=\pm \sqrt{k_d^2-q^2}$.

FIG. 16:

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

FIG. 17:

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

FIG. 18:

Effect of orientation on the emitter’s image. Top row: images of electric dipole emitters of fixed strength but different orientations in the $xz$-plane, where $\beta$ is the inclination angle; see Fig. 16. The emitter is situated 400 nm below the focal plane $(\text{NA}=1.2,n=1.33)$. Middle row: same as top row, but for the emitter situated in the focal plane. Bottom row: same again but for an emitter situated 400 nm above the focal plane. The scale bar is $0.5\mu \text{m}$.

FIG. 19:

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

FIG. 20:

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 results of the full wave-vector PSF calculation for an electric dipole emitter with fixed $x$-axis orientation, the blue curve the same calculation for a rapidly rotating (isotropic or random) emitter, the green curve presents the result of Eq. 71, and the ochre curve shows the Gaussian approximation of Eq. 74. Insets show two three-dimensional iso-surface PSF plots, left using the exact vector field calculation for an isotropic emitter, right for the scalar approximation. All calculations were performed for a water immersion objective with NA = 1.2.

FIG. 21:

Scalar approximation of the OTF of a wide-field microscope. Calculations were done for NA = 1.2 water immersion objective and an emission wavelength of 550 nm. The left panel shows the $k_x{k}_z$ cross-section of the electric field amplitudes in sample space, having a frequency support (frequencies with non-zero 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 $\Theta$. The middle panel shows the same distribution for the magnetic field. The right panel is the 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 color bar on the right hand side) normalized by the maximum absolute value of the corresponding amplitudes. For all panels, the coordinate origin $\left(k_x=0,k_z=0\right)$ is at the center. Throughout this review, we use the same representation for all OTFs shown.

FIG. 22:

Density plots of the first twelve Zernike polynomials as presented in table I: (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.

FIG. 23:

Model calculations of the image of an isotropic emitter (rapidly rotating dipole emitter) aberrated by a phase function given by the Zernike polynomials shown in Fig. 22. To better visualize the effects of aberration, all Zernike polynomials were multiplied by a factor 2.5. Calculations were again done for a water immersion objective with NA = 1.2 and for an emission wavelength of 550 nm. Yellow scale bar is $0.5\mu \text{m}$.

FIG. 24:

Total Internal Reflection Fluorescence (TIRF) microscopy. Excitation intensity above a coverslide interface with the sample medium as a function of incidence angle. The sample solution and coverslide refractive indices are, respectively, 1.33 (water) and 1.52, resulting in a TIR critical angle of ≈ 61°. The excitation wavelength is taken as 470 nm.

FIG. 25:

Super-critical Angle Fluorescence (SAF) microscopy. Ratio of super-critical to total downward fluorescence emission for a rapidly rotating molecule as a function of distance from the interface of the coverslide and the sample medium. The refractive indices of the sample solution and coverslide are, respectively, assumed to be 1.33 (water) and 1.52 (glass), with the emission wavelength of 550 nm. The inset shows the angular emission intensity distribution of an emitter directly on the interface (with the blue, red and green curves denoting UAF and SAF emissions, and emission towards sample solution, respectively). The SAF emission strongly depends on the emitter’s distance to the interface, while the under-critical emission is independent of emitter axial position. By determining the ratio of SAF to SAF+UAF emission, we can find the axial position of an emitter.

FIG. 26:

Metal-Induced Energy Transfer (MIET) microscopy: Dependence of the fluorescence lifetime (in terms of free space lifetime ${\tau }_0$) on the emitter’s distance from the glass substrate (coverslide) coated with a 20 nm gold layer. Calculations were done for an emission wavelength of 550 nm, and for a unit fluorescence quantum yield. Here we show the free curves for vertical, horizontal, and random emission dipole orientations. The inset illustrates the MIET sample geometry.

FIG. 27:

Geometry for deriving the electric field generated by a single dipole emitter above the MIET substrate (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 polar and inclination (azimuthal) angles, respectively. The three longer single-headed arrows show plane wave component vectors, with corresponding perpendicular polarization unit vectors ${\stackrel{\mathbf{ˆ}}{\mathbf{e}}}_{\parallel }$ and ${\stackrel{\mathbf{ˆ}}{\mathbf{e}}}_{\perp }^{\pm }$. Here ${\stackrel{\mathbf{ˆ}}{\mathbf{e}}}_{\perp }^{+}$ is the unit vector associated with the wave vector moving toward the metal surface. Similar conventions hold for the other unit vectors.

FIG. 28:

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

FIG. 29:

Schematic of the geometry of focusing a planar laser wavefront through the objective into the sample space; see Fig. 28. Wavefront patches at distance $\rho$ from the optical axis in the back focal plane are converted into spherical wavefront patches traveling at angle $\theta =\text{arcsin}(\rho /nf)$ with respect to the optical axis $z$, where $f$ is the focal length of the objective lens; see details in the main text.

FIG. 30:

CLSM and STED intensity distributions at the focus. Comparison of intensity distribution between conventional CLSM focus (left) with $z$-STED focus (middle) and $xy$-STED focus (right). Calculations were done for water immersion objective with NA = 1.2 at an excitation wavelength of 470 nm. On top of each column, the excitation polarization and its generating phase plate are shown. Bottom panels show 3D contour plots of the $1/e$, $1/e^2$ and $1/e^3$ intensity iso-surfaces and projections of $xy-,xz-$, and $yz$-cross-sections through the center.

FIG. 31:

Anatomy of the OTF (amplitude) of a confocal microscope. The left panel shows the excitation OTF. The middle panel shows the detection OTF for a confocal pinhole with $50\mu \text{m}$ radius and 60× magnification. The right panel shows the resulting confocal OTF obtained by a 3D convolution of the left two distributions.

FIG. 32:

OTF amplitude of a confocal microscope as a function of confocal aperture size. The confocal aperture radius is given at the top of each panel. Here, we assumed an excitation wavelength of 470 nm, emission wavelength of 550 nm, and a water immersion objective of NA = 1.2 at 60× magnification. The top most left panel shows the limit of an extremely large confocal pinhole so that 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 $(a=1\mu \text{m})$, so that the OTF approaches that of an ISM; see Sec. IV B 2.

FIG. 33:

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. 32 with 60× magnification.

FIG. 34:

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

FIG. 35:

Image formation in ISM. The blue curve represents the excitation intensity distribution $I_{\text{ex}}$ (excitation PSF) with its center at $\mathit{\xi }=0$ (optical axis). The yellow curve shows the detection PSF $\left(U_{\text{wf}}\right)$ for a pixel located at $\mathit{\xi }$ away from the optical axis. The pixel PSF $\left(U_{\text{pix}}\right)$, describing the image formation is, however, given by the product of the excitation and detection PSF, 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 $\mathit{\xi }/\kappa$.

FIG. 36:

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

FIG. 37:

4pi microscope excitation OTF generated by the interference of light focused through two opposing objectives. The left and middle panel 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 (excitation OTF). Excitation is assumed to be done using a water immersion objective with NA = 1.2.

FIG. 38:

Excitation PSF and (imaging) PSF of 4pi microscopy for a rapidly rotating emitter. The left panel shows the excitation PSF in the focus of a 4pi microscope, the middle panel shows the (imaging) PSF of a 4pi type A microscope, and the right panel that for a 4pi type C microscope. Calculations were performed using a water immersion objective with NA = 1.2 and 470 nm excitation wavelength and 550 nm fluorescence emission wavelength, and for a confocal detection in the limit of an infinitely small pinhole.

FIG. 39:

OTF of a type A 4pi microscope where excitation is done through two opposing objectives, and detection from one side through a confocal pinhole. For simplicity, we consider here only the limiting case of an infinitely small pinhole maximizing spatial resolution. The left panel shows the excitation OTF, the middle panel the OTF of detection with an infinitely small pinhole, and the right panel shows 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 NA = 1.2, and any Stokes shift between excitation and emission light is neglected.

FIG. 40:

OTF of a type C 4pi microscope. Similar to Fig. 39, 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. The left panel shows the excitation OTF, the middle panel the (identical) Fourier transform for coherent confocal detection from both sides, and the right panel shows the resulting OTF as a convolution of the two panels shown on the left.

FIG. 41:

Pixel reassignment in two-photon excitation ISM. By contrast to the ISM in Fig. 35, the excitation intensity distribution (one-photon excitation PSF) in two-photon microscopy has a larger width due to the larger excitation wavelength.

FIG. 42:

Comparison of one- and two-photon microscopy. For explanation see main text.

FIG. 43:

In (a) we show 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. In (b) we show a synthetic trace with 1500 photons generated assuming four molecules diffusing at $1\mu {\text{m}}^2/\text{s}$ for $30ms$ using background and molecule photon emission rates of ${10}^3\text{photons}/s$ and $4\times {10}^4\text{photons}/s$, respectively. The figure is adapted from Ref. [262].

FIG. 44:

Posteriors over diffusion coefficients strongly depend on the pre-specified $M$ when operating within a parametric Bayesian paradigm. The trace analyzed contains ≈1800 photons generated from 4 molecules diffusing at $D=1\mu {\text{m}}^2/s$ for $30ms$ with a background and maximum molecule photon emission rate of 10³ and $4\times {10}^4\text{photons}/\text{s}$, 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 panel d–and the mismatch in all others–highlights why we must use the available photons to simultaneously learn the number of molecules and $D$. The figure is adapted from Ref. [262].

FIG. 45:

Comparison of diffusion coefficients, $D$, obtained from the statistical framework versus FCS plotted against photon counts used in the analysis. Photon arrival times were simulated using the parameter values in Fig. 43b. The figure is adapted from Ref. [262].

FIG. 46:

Multi-focal setup uniquely resolving many molecular trajectories simultaneously. (a) A beam splitter is used to divide the fluorescent emission (designated by green) into two paths later coupled into fibers and detected by 4 APDs corresponding to different focal spots. (b) PSFs associated to different light paths. (c) Trajectories for two freely diffusing molecules with $D=1\mu {\text{m}}^2/s,{\mu }_0=5\times {10}^4$ photons/s and ${\mu }_{ℬ}={10}^3$ photons/s. Here, the orange and blue curves represent the learned trajectories’ ground truth and median, respectively. The blue and gray areas, respectively, denote the 95 percent confidence intervals and the PSF’s width. The figure is adapted from Ref. [261].

FIG. 47:

Lifetime histograms from single-pixel FLIM. Here, lifetimes are below the IRF and differ by subnanoseconds. Data sets used in panels (a-c) were simulated with 5⋅· 10², 10³, 2 · 10³ photons, IRF width of 0.66 ns, and ground truth lifetimes of 0.2 ns and 0.6 ns denoted by dotted lines. Learning the correct number of fluorophore species here requires > 500 photons.

FIG. 48:

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 sub-pixel interpolated lifetime maps obtained using the statistical framework of Box IV C. The average absolute difference between ground truth and learned maps is ≈ 4%. Scale bars are $4\mu \text{m}$. The figure is adapted from Ref. [53].

FIG. 49:

Sinusoidal illumination pattern for SIM microscopy. Here, ${\mathbf{k}}_i$ 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.

FIG. 50:

The SIM OTF. The left and middle panels, respectively, illustrate Fourier transforms of the modulated illumination intensity (SIM excitation OTF given by the three delta-peaks) and wide-field detection. The right panel shows the SIM OTF obtained by convolution of the two other panels; see also Eq. 123.

FIG. 51:

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 $\left({\text{OL}}_{\text{ex}}\right)$. Signal from the excited focal plane is collected through the detection objective lens $\left({\text{OL}}_{\text{det}}\right)$ and tube lens (TL) onto a camera (C). (b) In SPIM, a static light-sheet is formed by a cylindrical lens in the excitation path creating an elongated beam in one direction (above) and the same perpendicular detection optics as in panel a. (c) A schematic of the Gaussian beam in panels a-b focused through a lens or objective with diameter D, beam waist ${\omega }_0$ and Raleigh length $z_r$.

FIG. 52:

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

FIG. 53:

Multi-plane microscopy. (a) A conventional fluorescence microscope with epi-fluorescence (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. Here, the nominal focal plane is shown in black while the planes shown in red and blue can be also imaged by adjusting the axial positions of, for example, the sample. Shown are the sample (S), objective lens (OL), dichroic mirror (DM), and tube lens (YL). (b) A multi-plane microscope relays the optical path from the intermediate image formed in panel a via a telescope with lenses of focal lengths F₁, and F₂ 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, C2); see Ref. [350]. (c) A multi-focus microscope uses a multi-focus grating (MFG), i.e., diffractive element, chromatic correction grating (CCG) and prism (CCP) to achieve multiple focal planes on one camera; see text for more details.

FIG. 54:

Schematics for STED imaging. Excitation and depletion beams are used to acquire a sub-diffraction-limited image, formed after raster scanning the full sample. The resulting image can be understood as a convolution between the effective PSF combined from the excitation, and depletion laser beams, and the fluorescent molecule distribution in the sample. The image is adapted from Refs. [370, 371]. Schematics on the left hand side compare diffraction-limited confocal images of microtubules with the coinciding STED image. On the right panel we show the electronic transitions of excitation, and stimulated emission in STED (top), groundstate depletion GSD (middle), and RESOLFT (bottom). The figure is adapted from Ref. [377].

FIG. 55:

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

FIG. 56:

Single emitters are stochastically activated to become fluorescent. The activated emitters can be precisely localized provided they are spaced further apart than the Nyquist limit; see Sec. I C. 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 super-resolved pointillistic image reconstruction (see bottom panel right). Repetitive activation, localization, and deactivation temporally separate spatially unresolved structures in a reconstructed image with apparent resolution gain compared to the standard diffraction-limited image; see bottom row.

FIG. 57:

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 imager strand length. (c) Increasing either imager strand concentration or docking site density decreases dark times, ${\tau }_D$ (inter-event lifetime). The figure is adapted from Ref. [422].

FIG. 58:

PSF engineering. (a) Frequently used engineered PSFs, simulated for an objective lens with NA = 1.49 and pixel size of 110 nm. 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. I B) of the 3D position (each axis individually) plotted as a function of the axial position, assuming the system is laterally shift-invariant. Here, the subscripts in the axes labels indicate the coordinate for which CRLB was calculated.

FIG. 59:

A cartoon illustration of the CCD/EMCCD detector design detailed in the text.

FIG. 60:

A cartoon illustration of CMOS detector design detailed in the text.

FIG. 61:

Single photon detector. Laser pulses and their centers are, respectively, shown by blues spikes and red dashed lines with inter-pulse window $T$. The fluorophore excitation, photon emission and photon detection events take place, respectively, at $t_{\text{ext}},t_{\text{ems}}$ and $t_{\text{det}}$ designated by black dashed lines. The fluorophore spends time $\Delta t_{\text{ext}}$ in the excited state and emits a photon after $n$ pulses. The reported photon arrival time, $\Delta t_k$, is measured with respect to the immediate previous pulse center. Moreover, ${\Delta }_1$ and ${\Delta }_2$ denote the difference of the excitation pulse center and the detector delay in reporting the photon arrival time.