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Fluorescence Microscopy with Nanometer Resolution

Nanoscale Resolution in Far-Field Fluorescence Microscopy
  • Steffen J. SahlEmail author
  • Andreas Schönle
  • Stefan W. Hell
Chapter
Part of the Springer Handbooks book series (SHB)

Abstract

Throughout the twentieth century, it was widely accepted that a light microscope relying on propagating light waves and conventional optical lenses could not discern details that were much finer than about half the wavelength of light, or \(200{-}400\,{\mathrm{nm}}\), due to diffraction. However, in the 1990s, the potential for overcoming the diffraction barrier was realized, and microscopy concepts were defined that now resolve fluorescent features down to molecular dimensions. This chapter discusses the simple yet powerful principles that make it possible to neutralize the resolution-limiting role of diffraction in far-field fluorescence nanoscopy methods such as STED, RESOLFT, PALM/"​"​STORM, or PAINT. In a nutshell, feature molecules residing closer than the diffraction barrier are transferred to different (quantum) states, usually a bright fluorescent state and a dark state, so that they become discernible for a brief period of detection. With nanoscopy, the interior of transparent samples, such as living cells and tissues, can be imaged at the nanoscale. A fresh look at the foundations shows that an in-depth description of the basic principles spawns powerful new concepts. Although they differ in some aspects, these concepts harness a local intensity minimum (of a doughnut-shaped or a standing wave pattern) for determining the coordinate of the fluorophore(s) to be registered. Most strikingly, by using an intensity minimum of the excitation light to establish the fluorophore position, MINFLUX nanoscopy has obtained the ultimate (super)resolution: the size of a molecule (\(\approx{}{\mathrm{1}}\,{\mathrm{nm}}\)).

optical nanoscopy super-resolution microscopy single-molecule analysis biophysical imaging materials science 

The discovery of the diffraction barrier in 1873 by Ernst Abbe [22.1] showed that a far-field (i. e., focusing) light microscope could not resolve spatial structures that were smaller than, in the very best case, about a third of the wavelength of the focused light. For many decades, this physical insight had been accepted as a virtually unalterable limitation of focusing light microscopy and had consequently triggered the invention of nonoptical imaging techniques such as electron and scanning probe microscopy. In spite of the tremendous improvement in resolution brought about by these methods, light microscopy has maintained its importance in many fields of science. The reasons are mainly a number of exclusive advantages, the most prominent of which is the ability to almost noninvasively image (living) specimens. Light microscopy also provides the possibility to use fluorescence as a highly specific signature of the specimen features of interest. Fluorescence is particularly attractive when provided by endogenous fluorescence markers, e. g., by proteins in physiologically intact cells. Mapped with a confocal or multiphoton excitation microscope [22.2, 22.3, 22.4], fluorescence emission readily yields three-dimensional (3-D) distributions of proteins, or that of other fluorescence-labeled molecules from the strongly convoluted inside of biological specimens.

Abandoning the concept of focusing light altogether, near-field optical microscopy is the earliest practically relevant attempt to overcome the diffraction barrier with visible light [22.5]. To this end, scanning near-field optical microscopes employ ultrasharp tips or tiny apertures to confine the interaction of the light field with the object to sub-diffraction dimensions. However, applying this technique to soft biological matter has not been very successful, and it has to be applied carefully to avoid imaging artifacts [22.6]. In any case, a near-field optical microscope is confined to imaging surfaces, again underscoring the importance of improving the resolution in an optical microscope that still preserves focusing.

This formidable problem has been approached by many scientists [22.7, 22.8], but it was not until the mid-1990s that methods were described [22.10, 22.11, 22.9] that effectively broke Abbe's diffraction barrier , implying the potential to attain molecular resolution with regular lenses and visible light [22.12, 22.13, 22.14]. Over the past two and a half decades, with the development of several key experimental aspects and also aided by the increasing availability of suitable lasers and fluorophore labels, the concepts have fulfilled their original promise of nanometer-range resolution. The result has been a strong rise in research activity in nanoscopy, and recent breakthrough results indicate that the journey is by no means over. In this chapter, we initially trace the origins of diffraction-unlimited imaging with focused visible light. We discuss in detail the foundations of the field and the concepts underlying the achievement of nanoscale resolution in the optical far-field. The chapter then focuses on several important advances since the initial experimental demonstration of sub-diffraction resolution in 1999/2000. With the explosion of activity from the mid-2000s, we can cover merely a selection of this work, highlighting major steps and documenting the present state of the art. In particular, we limit ourselves to concepts and their implementations which have reached deeply sub-diffraction resolution, not just shifted or extended the long-standing resolution limit.

22.1 The Resolution Limit

The limited resolution of a focusing light microscope is readily described by the shape and extent of the effective focal spot, commonly referred to as the (effective) point spread function ( ). The PSF is the image that an infinitely small object would create. For an incoherent imaging mode such as fluorescence, the image is therefore given by the convolution of the object with the PSF
$$I=h\otimes G\;.$$
(22.1)
Here, \(I\), \(h\), and \(G\) denote the image, the PSF, and the object, respectively, with the object consisting of, for example, a distribution of fluorophores in space. The convolution actually means that the object is smeared out by the focal spot. While a careful analysis of the imaging properties requires a detailed analysis of the PSF, for PSFs with a single peak, assessing the full width at half maximum ( ) usually gives a good estimate of the microscope's resolution. If identical molecules are within the FWHM distance, the molecules cannot be separated in the image. Therefore, it becomes evident that improving the resolution is largely equivalent to narrowing the PSF of the microscope.
In a conventional fluorescence microscope, the FWHM of the PSF is about
$$\frac{\lambda}{2{n}\sin{\alpha}}=\frac{\lambda}{2\mathrm{NA}}\;,$$
with \({\lambda}\) denoting the wavelength, \(n\) the refractive index, \({\alpha}\) the semi-aperture angle, and \(\mathrm{NA}\) the numerical aperture of the lens. Without changes to the principal method, the spot size can be decreased only by using shorter wavelengths and larger aperture angles [22.1, 22.15]. However, the lens half-aperture is technically limited to \(\approx{}70^{\circ}\), and the wavelength \({\lambda}\) cannot be reduced below \({\mathrm{350}}\,{\mathrm{nm}}\), because shorter wavelengths are not compatible with live-cell imaging. In the best case, established far-field microscopes resolve \({\mathrm{180}}\,{\mathrm{nm}}\) in the focal plane (\(x,y\)) and merely \(500{-}800\,{\mathrm{nm}}\) along the optical axis (\(z\)) [22.16] (Fig. 22.1a-c). All attempts to improve the resolution merely by improving the optical components remain limited by diffraction. This is even better understood in the frequency domain. Here, the resolving power is described by the optical transfer function ( ), giving the strength with which these spatial frequencies are transferred from the object to the image. The OTF is readily computed as the Fourier transform of the PSF [22.17, 22.3]. Due to the convolution theorem, (22.1) becomes
$$\hat{I}=\hat{h}\hat{G}=o\hat{G}\;,$$
(22.2)
where the hat signifies a three-dimensional () Fourier transform, and \(o\) is the OTF. Because of the multiplication on the right-hand side, it is evident that even under optimal imaging conditions, object frequencies are lost when the OTF is zero. This loss is physically irrecoverable. Hence, the ultimate resolution limit is given by the highest frequency where the OTF is nonzero, i. e., the extent of the support of the OTF.
Fig. 22.1a-c

(a) The wave front created by a single lens is a spherical cap. For the highest available semi-aperture angle (\(\approx{}70^{\circ}\)), diffraction theory dictates an elongated spot along the optical axis (\(z\)) that is \(\approx{}3\) times as long as it is wide (\(xy\)). The elongation can also be interpreted by the fact that the wave front is only a cap rather than a sphere, bringing about an asymmetry of the focusing process. (b) Coherent addition of two spherical wave fronts created by opposing lenses completes a major part of the missing wave front towards a more spherical one, which is the tenet of the 4Pi concept. (c) The central focal spot, i. e., the main diffraction maximum, becomes more spherical as a result. However, because the wave front is not approaching a sphere, the main maximum is elongated in the lateral direction and side maxima appear along the optical axis

The highest spatial frequency produced by a (quasi-)monochromatic wave passing through the setup is \(k=2{\uppi}{n}/{\lambda}\). For focused light, the Fourier transform of its electric field, \(C(k)\), is therefore given by a spherical cap with radius \(k\), as seen in Fig. 22.2a-ca. The excitation probability is well approximated by the absolute square of the electric field, corresponding to an autocorrelation in frequency space. This signifies that frequencies of up to \(2k\) are contained in the excitation OTF. Nevertheless, resolving optics can be employed in both the illumination and the detection path. This introduces a spatially varying detection probability for the emitted fluorescence. Since the phase of the illuminating light is lost during excitation, fluorescence is emitted incoherently. Therefore, the effective PSF is given by the multiplication of the excitation PSF with its normalized detection counterpart, i. e., the detection PSF [22.3], which is equivalent to a convolution in frequency space. Thus the achievable cutoff is at \(2k_{\mathrm{ex}}+2k_{\mathrm{det}}\), where \(k_{\mathrm{ex}}\) and \(k_{\mathrm{det}}\) are the frequencies corresponding to the excitation and detection wavelengths in the medium, respectively.

Fig. 22.2a-c

OTF supports in frequency space. (a) For a single lens, the Fourier transform of the electric field is a spherical cap with a cutoff angle determined by the aperture angle of the objective lens. Excitation and detection (intensity) PSFs depend on the absolute square of the electric field; hence, the respective OTFs are given by the autocorrelation of the cap. Finally, in a confocal fluorescence microscope, the effective PSF is given by the product of the excitation and detection (intensity) PSF. Its OTF is therefore the convolution of the corresponding excitation and detection OTF, resulting in a squeezed OTF along the \(z\)-direction accounting for the lower axial resolution. (b) With a scalar field, a truly spherical wave front could ideally be generated. The autocorrelation of such a complete spherical shell is nonzero within a sphere of radius 2k. Convolution of the excitation and detection OTFs again results in a spherical OTF support of radius \(2k_{\mathrm{ex}}+2k_{\mathrm{det}}\), which is tantamount to isotropic resolution. This radius defines the theoretical resolution limit of all microscopes in which the resolution is created solely by the optical arrangement and in which the phase of light is lost at the sample, as in the excitation-emission process. (c) The 4Pi microscope of type C with coherent excitation and detection through both lenses with large aperture angles generates the largest wave front currently achievable in practice. For the highest available cutoff angles of \(\approx 70^{\circ}\), this results in an almost spherical effective OTF

These considerations are illustrated in Fig. 22.2a-c. As an example, if fluorescence is excited at \({\mathrm{488}}\,{\mathrm{nm}}\) and detected at around \({\mathrm{530}}\,{\mathrm{nm}}\), the cutoff is given by \((2n/{\mathrm{488}}\,{\mathrm{nm}}+2n/{\mathrm{530}}\,{\mathrm{nm}})^{-1}\cong{}{\mathrm{127}}\,{\mathrm{nm}}/n\), corresponding to approximately \({\mathrm{80}}\,{\mathrm{nm}}\) for oil and \({\mathrm{96}}\,{\mathrm{nm}}\) for water immersion. Therefore, all far-field light microscopes relying solely on improvements in the instrument have an ultimate resolution limit of about \({\mathrm{100}}\,{\mathrm{nm}}\) in all directions, even when all conceivable spatial frequencies are transmitted and restored in the image. This theoretical benchmark is indeed almost achieved by the 4Pi microscope described later in more detail [22.18, 22.19].

To overcome this fundamental limit, higher frequencies need to be generated. Obviously, this can be achieved by introducing nonlinearities in the interaction of the excitation light with a dye: For example, when using multiphoton (\(m\)-photon) excitation, the probability of producing a fluorescence photon depends on the \(m\)-th order of the illumination intensity [22.2, 22.20, 22.21]. The original excitation OTF will be convolved \(m\) times with itself, thus extending the support region to \(m\) times higher frequencies. Therefore, it has sometimes been argued that super-resolution can be attained by multiphoton excitation. It is readily understood, however, why the resolution limit cannot really be pushed [22.4]. While it is true that the excitation volume is narrower than the focal spot, which is due to the nonlinear dependence, using \(m\)-photon excitation to excite the same fluorophore also means that the excitation energy has to be split between the photons [22.4]. Consequently, the photons have an \(m\)-times lower amount of energy, and thus an \(m\) times longer wavelength \(m{\lambda}_{\mathrm{ex}}\), which results in \(m\) times larger focal spots to begin with. Consequently, the theoretical cutoff remains equal at \(2m(k_{\mathrm{ex}}/m)+2k_{\mathrm{det}}=2k_{\mathrm{ex}}+2k_{\mathrm{det}}\). The multiplication by \(m\) stems from the \(m\) correlations or convolutions in frequency space and the division from the longer excitation wavelength. Obviously, this holds equally for excitation with several photons of different energies, such as two-color two-photon excitation [22.22]. However, in the case of multiphoton excitation, higher frequencies within the support are usually damped. Multiphoton excitation is therefore not a viable option for obtaining a significant increase in the resolution. In addition, multiphoton excitation, especially for \(m> 3\), requires very high intensities [22.23], leading to complicated setups and damage to the sample. Bleaching is largely restricted to the focal plane, but there it can be much stronger than for one-photon excitation. For some dyes, however, the excitation wavelength can be chosen to be shorter than \(m{\lambda}_{\mathrm{ex}}\), and when imaging deep into scattering samples such as brain slices, multiphoton imaging can feature a superior signal-to-noise ratio ( ). In these cases, multiphoton excitation is a valuable method for reasons unrelated to the classical resolution issue.

Recognizing that the energy subdivision prevents any increase in resolution, multiphoton concepts have been proposed where the detection of a photon occurs only after the consecutive absorption of multiple excitation photons. Since the photons induce a linear optical transition in the dye, high intensities are not required and photon-energy subdivision does not occur [22.24, 22.25, 22.26]. However, these concepts require specific conditions or complicated dye systems, also complicating their practical realization.

A radically different concept for improving far-field fluorescence microscopy resolution appeared in the mid-1990s in the form of stimulated emission depletion ( ) and ground state depletion ( ) microscopy [22.10, 22.9]. Both concepts were based on one common principle: a spatially modulated reversible saturable optically linear (fluorescence) transition ( ) between two molecular states allows the reversible transfer of fluorophores between (at least) two states, typically a bright fluorescent state and a dark (i. e., non-signaling) state. This introduces the ability to render fluorophores even at deeply sub-diffraction length scales discernible for the brief period of detection. Since there is no limit to the confinement of the fluorophore state, there is no longer a fundamental limit to the resolution other than the size of the fluorophore itself, as the fluorophore is the smallest separable entity. Therefore, concepts based on the RESOLFT principle, such as STED or GSD microscopy, truly break the diffraction barrier. Because of their principles of operation, they can achieve nanoscale resolution in all directions [22.11, 22.12, 22.27]. Fundamentally departing from earlier approaches to increased resolution, an introduction to the general idea will follow later in the chapter, where we also outline different possibilities for realizing the RESOLFT concept. Following the first demonstration of a spatial resolution of \({\lambda}/25\) with focused light using regular lenses [22.28], the concepts, and in particular STED fluorescence imaging, have since been developed into versatile and widely applicable methods.

22.2 Improvement in Axial Resolution by Aperture Enlargement:4Pi Microscopy and Related Approaches

Figure 22.2a-c illustrates that the suboptimal axial resolution of a far-field light microscope can be readily motivated by the fact that the focusing angle of the objective lens is far from covering a full solid angle of \(4{\uppi}\). The axially elongated PSF evidently also implies an axially compressed OTF. If the focused wave fronts were virtually spherical, the focal spot, as well as the OTF of the system, would be as well. In this case, the resolution would be largely isotropic. Therefore, an obvious way to achieve optimal optical resolution is to synthesize a focusing angle that comes closer to the solid angle of \(4{\uppi}\). Since the focusing angle of a lens is limited by technical constraints, a larger wave front can be obtained only by pasting together two or more wave fronts. A wave-front synthesis sharpening the focus along the optical axis is attained by employing two opposing lenses (Fig. 22.1a-c) that either coherently illuminate the sample from both sides, or detect the light through both lenses in a coherent manner [22.29, 22.30, 22.31]. The resulting support of the OTF is displayed in Fig. 22.2a-cc. In the spatial domain, the effect can be pictured as two counter-propagating focused beams interfering either at the focal spot or at a common point at the detector for the illumination and the detection, respectively. For the common high-angle lenses with a semi-aperture \(64^{\circ}<\alpha<72^{\circ}\), constructive interference produces a maximum with \(\approx{}3{-}4\) times narrower axial FWHM. The basic idea of the 4Pi microscope is to produce and take advantage of this axially narrowed PSF. Because the semi-aperture angle \(\alpha\) is considerably smaller than \(90^{\circ}\), the main maximum is also accompanied by \(1{-}2\) axial satellite lobes originating from the missing lateral part of a potentially full solid angle (Fig. 22.1a-cc).

Importantly, the narrowed main focal maximum is not \({\lambda}/(4n)\), as one would anticipate for a flat standing wave, but is somewhat larger. By the same token, the side maxima are not located at \(m{\lambda}/(2n),(m=0,1,2,3,\ldots{})\), but slightly farther away from the focal point, constantly decreasing in height with increasing order. As we shall see further down, this difference is essential to the improvement in axial resolution brought about by the coherent use of two opposing lenses, and is the actual reason that it is not possible to improve the axial resolution just by the interference of counter-propagating plane waves [22.32, 22.33].

The latter is the tenet of the standing wave microscope ( ), which utilizes flat standing waves of laser light and thus a set of excitation nodal planes in the sample along the optical axis. The SWM uses wide-field detection on a camera, like any conventional fluorescence microscope; the flat standing wave of excitation light is produced either with a mirror beneath the sample or by adding counter-propagating waves from two lenses. A substantial increase in axial resolution had initially been claimed by SWM. However, the resulting multiple thin interference layers in a sample [22.32] do not unambiguously resolve features extending over a minimum axial range that is larger than half the wavelength (\(0.2{-}0.4\,{\mathrm{{\upmu}m}}\)) [22.34, 22.35]. While SWM might prove useful for specialized applications, it fails in delivering axial images of arbitrary objects [22.36, 22.37]. One explanation for this is that the interference excitation maxima have virtually equal intensity, leading to strongly blurred ghost images. The deeper-rooted physical reason is that SWM does not increase the aperture of the system. This can be readily observed when analyzing the SWM in the frequency space (Fig. 22.3). The excitation OTF has only three delta peaks located at \(-k,0\), and \(k\) on the inverse optical axis. When convolved with the detection OTF, large gaps remain in the support of the effective OTF representing potential object frequencies that are not transferred from the object to the image.

Fig. 22.3

Comparison of the supports of excitation, detection, and effective OTFs for techniques using two opposing lenses coherently. The effective OTF is the convolution of the excitation and detection OTFs. For two-photon excitation in the 4Pi mode, the Fourier transform of the excitation intensity is given by the excitation OTF of the 4Pi type C microscope but scaled by \({\mathrm{488}}\,{\mathrm{nm}}/{\mathrm{800}}\,{\mathrm{nm}}=0.61\). The excitation OTF is therefore very similar to the effective OTF of 4Pi type C. Note the gaps in the support of the SWM microscope's effective OTF

It is a major physical insight (into the problem of axial resolution improvement with coherently used opposing lenses) that these gaps can only be closed when the light is focused, that is with spherical wave fronts. Focusing to the same point requires the accurate alignment of the two lenses, but the real physical challenge is to identify feasible mechanisms helping to further reduce the fluorescence contributed by the side-maxima still present when focusing at \(64^{\circ}<\alpha<72^{\circ}\). Reduction of the contribution from the side-maxima is equivalent to completing the support of the OTF and eventually to avoiding imaging artifacts. When spot-scanning confocal and non-confocal 4Pi microscopy [22.29, 22.38] (Fig. 22.3) and then wide-field I5M microscopy [22.31] finally demonstrated the improvement of axial resolution [22.19, 22.39, 22.40, 22.41], each method used lenses with high numerical aperture and relied on at least one of the following three lobe-reducing mechanisms:
  1. i)

    The use of both focused excitation and confocal detection [22.29, 22.30] suppresses fluorescence from higher-order side lobes of the excitation PSF.

     
  2. ii)

    Two-photon excitation ( ) [22.38] emphasizes differences in the intensities of the main and the side maxima due to its quadratic dependence on the excitation intensity.

     
  3. iii)

    Finally, disparities between excitation and fluorescence wavelengths can be used [22.30, 22.31, 22.38] if light is detected coherently through both lenses. This is explained by the fact that the intensity maxima for excitation and detection are located at different points in space due to the large difference in wavelength between fluorescence excitation and emission (Stokes shift).

     

Three major types of 4Pi microscopy have been reported [22.30]. They differ in whether the spherical wave fronts are coherently added for illumination, for detection, or for both simultaneously; they are referred to as types A, B, and C, respectively. Typically the detection has been confocalized, but in conjunction with TPE, successful axial separation with non-confocal detection has also been reported. Here we will concentrate on the TPE 4Pi (type A), the 4Pi type C, and the TPE 4Pi type C confocal microscopes. Of these three, the TPE 4Pi-confocal microscope has been applied to the largest number of imaging problems. It uses the very effective lobe-reducing measure of TPE combined with point-like detection. In reality, the size of the point-like detector amounts to about the size of the main maximum of the diffraction-limited fluorescence spot (Airy disk) , when imaged into the focal plane of the objective lens.

Clearly, non-confocal wide-field detection and regular illumination would make 4Pi microscopy more versatile. Therefore, the related approach of I5M [22.31, 22.40, 22.41, 22.42] confines itself to using the simultaneous interference of both the excitation and the (Stokes-shifted) fluorescence wave-front pairs; the latter are spherical as in a 4Pi microscope.

The potential benefits of I5M are readily stated: single-photon excitation with arguably less photobleaching, an additional \(20{-}50\%\) gain in fluorescence signal, and lower cost. This method has yielded 3-D-images of actin filaments with an axial resolution slightly better than \({\mathrm{100}}\,{\mathrm{nm}}\) in fixed cells [22.41]. To remove the side-lobe artifacts, I5M-recorded data have been deconvolved offline. While the consideration of the OTF support in Fig. 22.3 suggests that this single mechanism is indeed sufficient, it turns out that the relaxation of the side-lobe suppression comes at the expense of increased vulnerability to sample-induced aberrations, especially with non-sparse objects [22.36, 22.37]. Thus I5M imaging, which to date has relied on oil immersion lenses, has required mounting the cell in a medium with \(n=1.5\) [22.41]. Live cells inevitably necessitate aqueous media (\(n=1.34\)). Moreover, water immersion lenses have an inferior focusing angle and therefore larger lobes to begin with [22.43]. Potential strategies for improving the tolerance of I5M are the implementation of a nonlinear excitation mode and its combination with pseudo-confocal or patterned illumination [22.44]. While these measures again add physical complexity, they may have the potential to render I5M more suitable for imaging in living cells.

The implementation of at least two of the mechanisms above has proved fairly reliable: After initial demonstration of TPE 4Pi-confocal microscopy [22.39], super-resolved axial separation was applied to fixed cells [22.19]. The image quality was further improved by applying nonlinear restoration [22.45, 22.46, 22.47]. Under biological imaging conditions, this typically improves the resolution up to a factor of \(\mathrm{2}\) in both the transverse and axial directions. Therefore, in combination with image restoration, TPE 4Pi-confocal microscopy has resulted in a resolution of \(\approx{}{\mathrm{100}}\,{\mathrm{nm}}\) in all directions, as first witnessed by the imaging of filamentous actin [22.19] and immunofluorescence-labeled microtubules [22.48, 22.49] in mouse fibroblasts.

While a very useful and explanatory comparison of the OTF supports has been published [22.40], it is obvious from the above paragraph that consideration of the supports alone is not sufficient to understand the respective benefits and limitations of SWM, I5M, and 4Pi microscopy. Rather than comparing their technical implementation [22.41], a quantitative analysis of their PSFs and OTFs is needed to clarify under which conditions these microscopes will be able to deliver 3-D-resolved images with superior resolution. The success of increasing the axial resolution with coherently used opposing lenses depends not only on the achievable bandwidth, but also on the strength with which the respective systems transfer the spatial frequencies within this bandwidth. Gaps and weak parts that occur in some systems [22.31, 22.34] must be quantified for a particular optical setting because they may render the removal of artifacts impossible, making an increase in axial resolution impossible. Below we demonstrate that these gaps are intimately connected with the optical arrangement and therefore inherent to some of the methods described.

22.2.1 The Optical Transfer Function of 4Pi Microscopy and Related Systems

Before expanding on the analysis of PSFs and OTFs, we quickly review their theoretical derivation. The excitation PSF of the confocal microscope, as well as the detection PSF of the wide-field, confocal, SWM, and 4Pi type microscopes, is a regular-intensity PSF. Assuming that excitation and emission involve a dipole transition of the dye, the excitation and emission PSF of a single lens is well approximated by the absolute square of the electric field in the focal region
$$h=\left|\boldsymbol{E}_{1}(z,r,\phi)\right|^{2}.$$
(22.3)
It depends on the axial and radial distance to the geometrical focus (\(z\) and \(r\), respectively) and the polar angle \({\phi}\). The field is usually calculated using the vectorial theory of Richards and Wolf [22.50]. In our case, we assumed circular polarization of the light. In frequency space, the Fourier transform of the electrical field is then simply given by a spherical cap of radius \(k_{\mathrm{ex}}\) (Fig. 22.2a-c). This is equivalent to approximating the spherical wave fronts emerging at the exit pupil as plane waves close to the focal spot. The absolute square in (22.3) corresponds to an autocorrelation in frequency space. When calculating the OTF, this convolution can be carried out directly [22.51, 22.52], or the OTF can be determined by Fourier transformation of (22.3).
For the excitation PSF of the 4Pi-confocal microscopes (types A and C) and for the detection PSF of the I5M and the 4Pi-confocal microscopes (types B and C), a calculation of constructive interference between the two spherical wave fronts is required. The PSF is therefore given by the coherent addition of two beams
$$h=\left|\boldsymbol{E}_{1}(z,r,\phi)+\boldsymbol{E}_{2}(z,r,\phi)\right|^{2}.$$
(22.4)
The field of the opposing lens is given by
$$\boldsymbol{E}_{2}(\mathbf{r})=\mathbf{M}\boldsymbol{E}_{1}(\mathbf{M}^{-1}\boldsymbol{r})\;.$$
(22.5)
\(\mathbf{M}\) is the coordinate transform from the system of lens number 2 to lens number 1 and is a diagonal matrix inverting the \(z\)-components for a triangular cavity and the \(y\)- and \(z\)-components for a rectangular cavity [22.53]. In frequency space, the Fourier transform of the electric field is now given by two caps corresponding to the two focused wave fronts. Consequently, the OTF consists of an autocorrelation part equivalent to that for single-lens excitation or detection and a cross-correlation of two opposite spherical caps represented by the outer brackets in Figs. 22.2a-c22.4. For TPE, the excitation PSF is simply given by squaring the one-photon PSF scaled to the appropriate TPE wavelength, and similarly, the OTF is the auto-convolution of the scaled one-photon OTF [22.4].
The excitation intensity of the SWM is given by a plane standing wave along the optical axis,
$$h=I_{0}\cos^{2}\left(kz\right),$$
(22.6)
where \(I_{0}\) denotes a constant, \(k\) is the wave number, and we assumed constructive interference to occur at the common geometrical focus. The excitation OTF is its Fourier transform and is given by
$$o=\frac{I_{0}\left[\delta(k)+\frac{\delta\left(k-2k_{\mathrm{ex}}\right)}{2}+\frac{\delta\left(k+2k_{\mathrm{ex}}\right)}{2}\right]}{2}\;.$$
(22.7)
Again, this is the result of autocorrelating the electric field's Fourier transform that consists of delta peaks at \({\pm}2k_{\mathrm{ex}}\).
While the 4Pi microscope uses a spatially coherent point-like laser illumination, in the I5M microscope, wide-field illumination is used, normally in the Köhler mode, with either a lamp or a laser. The physical consequences of this difference are best explained as follows. If one decomposes the two spherical wave fronts of the 4Pi illumination into plane waves incident at different angles and corresponding to different points of the illumination apertures, all plane waves of the aperture interfere with each other in the focal region. In the I5M, the illumination light is not coherent throughout the aperture. Therefore, only pairs of plane waves originating from corresponding points (mirror images about the focal plane) of the illumination apertures are mutually coherent, forming a standing wave in the focal region. The period of these standing waves scales with the cosine of the azimuth angle \({\theta}\). The excitation PSF of the I5M can then be calculated by adding the intensity of these plane standing waves. Assuming uniform intensity throughout the exit pupil of the lens, the PSF is given by
$$\begin{aligned}\displaystyle h(z)&\displaystyle=I_{0}\int\mathrm{d}\phi\int_{0}^{\alpha}\mathrm{d}\theta\,\sin\theta\cos^{2}(k_{\mathrm{ex}}z\,\cos\theta)\\ \displaystyle&\displaystyle=2\uppi I_{0}\int_{0}^{\alpha}\mathrm{d}\theta\,\sin\theta\cos^{2}(k_{\mathrm{ex}}z\,\cos\theta)\;.\end{aligned} $$
(22.8)
The support of the excitation OTF is readily inferred: For each \({\theta}\), the integrand in (22.8) is the excitation PSF of the SWM, and thus the total OTF consists of a superposition of expressions of (22.6) for wave vectors ranging from \(k_{\mathrm{ex}}\cos({\alpha})\) to \(k_{\mathrm{ex}}\). Loosely speaking, this incoherent superposition smears out the delta peaks at the sides, forming the lines in Figs. 22.3 and 22.4. This excitation mode contributes to avoiding the gaps that remain in the SWM's support after convolution with the detection OTF. If, on the other hand, the incoherent light source is imaged into the focal plane of the lens, i. e., critical illumination, mutually coherent points form wave fronts focused onto and interfering at the conjugate point in the image of the light source. The individual 4Pi PSFs produced by each point of the light source as a result are incoherently summed up, giving an integral of the 4Pi excitation PSF over the field of view in the focal plane. The OTF becomes nonzero exclusively on the inverse optical axis, where it is given by the values of the 4Pi OTF, altogether leading to a result not very different from that predicted by (22.8). However, critical illumination is problematic due to potential nonuniformity in the light source and will be omitted in our analysis.

In any case, once the fluorescence light is generated in the sample, the I5M collects the spherical fluorescent wave fronts just as in a 4Pi microscope of type B or C. The two counter-propagating spherical wave fronts of fluorescence collected by each lens interfere constructively in a common point on the camera. Disturbance of the interference pattern of neighboring points of fluorescence emission is damped by the spatial incoherence in the focal plane: the radius of spatial coherence is largely given by the Airy disk associated with the fluorescence light at the aperture in use. Thus the I5M implements the highest possible degree of parallelization of 4Pi detection.

Figure 22.4 shows the numerically calculated excitation, detection, and effective PSFs/OTFs of the 4Pi, I5M, and SWM setups, along with those of the conventional epifluorescence and confocal microscope. The epifluorescence microscope features uniform illumination intensity throughout the sample volume; its OTF is a single delta peak at the origin. To obtain a practically relevant comparison, we assumed a numerical aperture of \(\mathrm{NA}=1.35\) and oil immersion with a refractive index \(n=1.51\). In the case of single-photon excitation of the dye, an excitation and detection (i. e., central fluorescence) wavelength of \(\mathrm{488}\) and \({\mathrm{530}}\,{\mathrm{nm}}\), respectively, was assumed. For TPE, an excitation wavelength of \({\mathrm{800}}\,{\mathrm{nm}}\) was chosen. Finite-sized pinholes can be taken into account by convolving the detection PSF with the image of the pinholes in the focal plane: a disk of radius \(r_{\mathrm{PH}}\). In frequency space this corresponds to a multiplication with the disk's Fourier transform given by
$$\hat{h}_{\mathrm{PH}}=\frac{J_{1}(kr_{\mathrm{PH}})}{k}\;.$$
(22.9)
The first root of the Bessel function is at \(\approx{}3.83\), and thus the detection OTF becomes zero at \(k=3.83/r_{\mathrm{PH}}\). However, the largest frequency present in the detection OTF is given by half the wavelength, and therefore its support is unaltered if the pinhole radius is smaller than \(3.83{\lambda}/(4{\uppi}\mathrm{NA})\cong 0.3{\lambda}/\mathrm{NA}\), which corresponds to half the Airy disk radius. Pinholes smaller than this can be neglected, while for sizes around this value and larger, the PSF will widen laterally, suppressing the OTF at higher lateral frequencies. The effect of the pinhole size on axial resolution remains small as long as it does not exceed that of the Airy disk. We will therefore neglect the pinhole in our further analysis. All PSFs were numerically computed in a volume of \(128{\times}128{\times}512\) pixels in \(x\)-, \(y\)-, and \(z\)-directions, respectively, for cubic pixels with \({\mathrm{20}}\,{\mathrm{nm}}\) length. The OTFs were calculated by Fourier transformation; of the 512 pixels in \(z\)-direction, only data based on the central 256 pixels are shown in Fig. 22.4. The color lookup table (LUT ) has been chosen so that the regions of weak signal are emphasized for both PSF and OTF. This reveals important differences between the systems. Areas of low but non-negligible intensity are important, since they cover a large volume and contribute substantially to the image formation.
Fig. 22.4

Overview of the excitation, detection, and effective OTFs' modulus and corresponding PSFs of the wide-field, confocal, standing wave (SWM), I5M, 4Pi-confocal type C, TPE 4Pi-confocal type A, and TPE 4Pi-confocal type C microscopes. The color lookup table (LUT) has been designed to emphasize the important weak OTF regions. The OTFs are shown in the squares above the corresponding PSFs; the zero-frequency point is in the center, and the largest frequency displayed is \(2{\uppi}/{\mathrm{80}}\,{\mathrm{nm}}^{-1}\). The circles represent the maximum possible carrier. For TPE, the excitation OTFs extend slightly over these circles because the excitation wavelength of \({\mathrm{800}}\,{\mathrm{nm}}\) is less than double the one-photon excitation wavelength of \({\mathrm{488}}\,{\mathrm{nm}}\). While all these methods extend the OTF along the axial direction, they differ fundamentally in contiguity and absolute strength within the support region

Let us first consider the PSFs (Fig. 22.4, narrow columns). Immediately, some differences between the various approaches become apparent. While the excitation modes of the SWM and I5M are similar, the local minima are not zero for the latter due to the incoherent addition of the standing wave spectrum. The most important difference is observed when comparing the 4Pi microscopes. As a result of focusing, their PSFs are confined in the lateral direction so that contributions from the outer lateral parts of the focal region are reduced. The confinement has important consequences. Whereas the 4Pi-confocal microscope, especially its two-photon version, exhibits only two pronounced but rather low lobes, the I5M and, even more so, the SWM feature a multitude of lobes and fringes on either side of the focal plane, despite the fact that all of them rely on the same aperture. The second consequence is that, due to its quadratic or cubic dependence on the excitation distributions, the 4Pi-confocal PSFs can be separated into an axial and a radial function in good approximation [22.54, 22.55]
$$h(r,z)\cong c(r)h_{\mathrm{l}}(z)\;.$$
(22.10)
Separability is a distinct feature of the 4Pi-confocal and multiphoton arrangements, and we shall later see that it is the prerequisite for simple online removal of side-lobe effects in the image. TPE leads to a further suppression of the outer parts of the excitation focus and thus of the side lobes of the 4Pi-illumination mode. Figures 22.3 and 22.4 also reveal that, in conjunction with coherent detection (type C), TPE 4Pi-confocal microscopy features an almost lobe-free PSF. Its OTF nearly fills the maximum support region. In the SWM and the I5M, the number and relative heights of the lobes increase dramatically when moving away from the focal point because an effective suppression mechanism is missing. In the SWM, the lobes become even higher than the central peak itself. In the I5M, the secondary maxima are as high as the first maxima of the single-photon 4Pi-confocal microscope of type C.

22.2.2 Removing Periodic ArtifactsThrough Deconvolution

Even if they are small, side lobes and resulting ghost images in the raw data remain a common feature of all methods that employ two lenses coherently. Next, we turn to the OTF in order to understand the circumstances under which image processing can be used to remove the artifacts induced by the lobes. It is obvious from (22.2) that if the OTF were nonzero everywhere, we could divide the Fourier transform of the image by it. Subsequent Fourier back-transformation of the data would render the object. In practice, the OTF is limited in bandwidth and has weak regions. As division outside the OTF support is impossible, these frequencies are lost. But even in regions where the OTF is small, division strongly amplifies noise, producing artifacts.

Iterative image restoration techniques, whether linear or nonlinear, aim to restore as many frequencies as possible while trying to avoid this effect. Linear deconvolution is based on the division approach but introduces a special treatment for frequencies not transmitted by the OTF. It is only capable of restoring frequencies where the OTF is above the noise level. If frequencies are missing, a correct representation of the object in the image can be given only if these frequencies are extracted from a priori knowledge of the object. This extraction is mathematically more complex and often not viable. Linear deconvolution, on the other hand, is computationally facile and fast. Speed is of particular importance because the interference artifacts are ideally removed online, making the final image readily accessible. Therefore, one of the most prominent advantages of a uniformly strong OTF is the ability to apply a linear deconvolution.

The comparison of the effective OTFs in Fig. 22.4 highlights the severe gaps in the SWM, making deconvolution impossible. Linear deconvolution is reportedly possible in the I5M [22.41]. However, as the gaps in the I5M are filled with rather low amplitudes, this method will create image artifacts for objects that are not sparse or not very bright, or that contain spatial frequencies coinciding with the gap. Owing to the contiguity of its support and the strong amplitudes of the OTF, 4Pi-confocal microscopy fulfills the preconditions for linear deconvolution. In fact, linear deconvolution along the axial direction based on the separability of the PSF has been applied for the removal of interference artifacts arising in the recording of complex objects. Thus super-resolved axial imaging has been shown in the dense filamentous actin [22.55] and in the microtubular network [22.48] of a mouse fibroblast cell. We will have a closer look at this important procedure in the following section.

Approximating the 4Pi-confocal PSF as in (22.10) allows us to perform a computationally inexpensive one-dimensional linear deconvolution that simply eliminates the effect of the lobes in the image. The axial factor of the PSF can basically be decomposed into the convolution of a function \(h_{\mathrm{p}}(z)\) describing the shape of a single peak, and a lobe function \({l}(z)\) containing the position and relative heights of the lobes
$$h_{\mathrm{l}}(z)\cong h_{\mathrm{p}}(z)\otimes l(z)\;.$$
(22.11)
\(h_{\mathrm{p}}\) quantifies the blur and \(l\) describes the replication that is responsible for the ghost images in unprocessed 4Pi images. The effect of the lobe function can be eliminated using algebraic inversion. We use a discrete notation where each lobe is represented by a component \(l_{i}\) of the vector \(\boldsymbol{l}\), with \(l_{0}\) denoting the strength of the central lobe and the index running from \(-n\) to \(n\). Negative indices denote lobes to the left of the central peak. If the lobe distance in pixels is denoted by \(d\), the values of the object along the line are given by \(O_{j}\) and those of the image by \(I_{j}\). Thus, the convolution is given by
$$I_{j}=\sum_{k}l_{k}O_{j-{d}k}\;.$$
(22.12)
Looking for a filter \(\boldsymbol{l}^{-1}\) inverting this convolution we need
$$O_{j}=\sum_{s}l_{s}^{-1}I_{j-{d}s}=\sum_{s}l_{s}^{-1}l_{k}O_{j-{d}s-{d}k}\;,$$
(22.13)
for all possible objects. The inverse filter needs to fulfill the condition
$$\sum_{s}l_{s}^{-1}l_{j-s}=\delta_{j0}\;.$$
(22.14)
At this point we can arbitrarily choose the length of the inverse filter, assuming an index running from \(-m\) to \(m\). The index \(j\) in (22.13) can take values from \(-m-n\) to \(m+n\). Therefore, we have a system of \(2(m+n)+1\) equations with \(2m+1\) unknowns, which is usually not solvable. An approximation is found by considering the equations for \(j=-m,{\ldots},m\) only. Now, the problem is equivalent to solving a linear Toeplitz problem with the Toeplitz matrix given by the vector \(\boldsymbol{l}\) [22.36, 22.56]. The approximation is good, if the edges of the inverse filters are small, since the remaining equations are nearly satisfied. This holds if the first-order lobes are \(<{\mathrm{45}}\%\); in this case, the error is practically not observable. A typical length of the inverse filter is 11. In conjunction with a lobe height of \({\mathrm{45}}\%\), the edges of the inverse filter feature a modulus below \({\mathrm{1}}\%\) of the filter maximum. For lobes of \({\mathrm{35}}\%\) relative height, this drops to a value of only about \({\mathrm{0.08}}\%\). Thus, using this technique, the separability of the 4Pi-confocal PSF allows one to swiftly obtain a final image that is equivalent to imaging with an ideal optical microscope that has a single narrow main maximum at the focal point. The inverse filter is discrete and nonzero only at a few points. This very effective side lobe removal method is therefore referred to as point deconvolution. Exploiting the characteristics of the PSF of the 4Pi microscope, it is applicable only in conjunction with this method.
Axial lobes in the PSF entail suppressed OTF regions along the optical axis. Thus, point deconvolution restores suppressed frequencies by linear deconvolution even though the actual calculation takes place in real space. For PSFs which cannot be written as in (22.10), point deconvolution is insufficient, because the transverse directions (\(x\),\(y\)) have to be involved as well. In this case, it is mathematically clearly preferable to pass the data through the frequency domain. Established linear deconvolution algorithms rely on inverting (22.2) but also account for the vanishing regions of the OTF. An estimate of the frequency spectrum of the object can be obtained by using
$$\hat{E}(k)=\hat{I}(k)\frac{o^{*}(k)}{|o(k)|^{2}+\mu}\;.$$
(22.15)
The regularization parameter \({\mu}\) [22.57] sets a lower threshold on the denominator to avoid amplification of frequencies where the modulus of the OTF is so small that the frequency spectrum of the image is dominated by noise. If the OTF is a convex function that is in the absence of gaps, the effect of regularization is similar to smoothing. In most cases considered in this chapter, however, the OTF is not convex and the situation is more complicated. Small OTF values are found not only at the OTF boundaries, but also within its region of support, e. g., in the vicinity of the minima (Fig. 22.4) or at the frequency gaps (Fig. 22.3). If the lobes are too high (typically \(> {\mathrm{50}}\%\)), the level of the minima of the OTF is comparable to or smaller than the noise level. This is definitely the case in the SWM, but also in the presence of slight aberrations in the I5M, as well as in an aberrated 4Pi microscope. Therefore, the necessity of implementing a certain value of \({\mu}\) is tightly connected with the lobe height and with the potential artifacts induced by the lobes.
If the PSF is separable in a peak function and an axial lobe function, the lobe removal can be elegantly targeted
$$h(\boldsymbol{r})=h_{\mathrm{p}}(\boldsymbol{r})\otimes l(z)\;.$$
(22.16)
Contrary to the decomposition in (22.11), the peak function \(h_{\mathrm{p}}(\boldsymbol{r})\) also contains the dependence of the peak on the lateral coordinates. We no longer require the PSF to separate into a radial and an axial part, yet the lobe function still gives the relative height of the lobes as well as their location
$$l\left(z\right)=\sum_{s}l_{s}\delta(z-ds)\;,$$
(22.17)
with \(d\) now being the lobe distance in units of length. The frequency spectrum of the image is then given by
$$\hat{I}(\boldsymbol{k})=\hat{G}(\boldsymbol{k})\cdot{}o(\boldsymbol{k})=\hat{G}(\boldsymbol{k})\cdot\hat{h}_{\mathrm{p}}(\boldsymbol{k})\cdot\hat{l}(k_{z})\;.$$
(22.18)
If the lobe function is symmetric with respect to the focal plane (i. e., for constructive interference), we can write
$$\hat{l}(k_{z})=1+\sum_{s> 0}2l_{s}\cos(sdk_{z})\;.$$
(22.19)
This decomposition immediately reveals why the critical lobe height is \({\mathrm{50}}\%\): If the PSF consists of a main maximum and two primary lobes of \({\mathrm{50}}\%\), the right-hand side of (22.19) vanishes for the axial frequencies associated with the distance d. So, if the lobes \(> {\mathrm{50}}\%\), the frequency represented by the lobes is not contained in the OTF, and hence not transferred to the image. In reality, the critical lobe height is slightly above \({\mathrm{50}}\%\). This is because of the influence of the secondary lobes that have been neglected in our reasoning. Nonetheless, the \({\mathrm{50}}\%\) threshold is an excellent rule of thumb for the critical lobe height, which applies equally to SWM, I5M, and 4Pi microscopy, for fundamental reasons.

Equation (22.18) implies that the effect of the lobes can be removed by direct Fourier inversion. This is the case if the Fourier transform of the lobe function remains above the noise level throughout the relevant frequency spectrum. The spectrum of the lobe-free image is then obtained by dividing the image spectrum by the Fourier transform of the lobe function. Figure 22.5a-f shows a typical data set acquired with a TPE type A 4Pi-confocal microscope and illustrates lobe removal in both the spatial and frequency domains. The avalanche photodiode used as a detector had a typical dark count rate \(<1\) count/pixel, which is negligible [22.19]. Hence, the only significant source of noise was the Poisson noise of the photon-counting process, manifesting itself as white noise that is independent of the spatial frequency. Since the primary lobes are well below \({\mathrm{50}}\%\), the first minima of the OTF are at \({\mathrm{19}}\%\), which is well above the typical noise level of \(0.5{-}1\%\). Direct lobe removal is straightforward in this example, underscoring the importance of non-vanishing amplitudes in the OTF. If the OTF exhibited regions close to zero, as is predicted for the I5M, the multiplication with a high number in this region would result in a strong amplification of noise, and would compromise the obtained image. In the SWM, it is impossible, on a general basis, to remove the interference artifacts.

Fig. 22.5a-f

Lobe removal and deconvolution in 4Pi microscopy. The same pair of actin fibers in a fixed mouse fibroblast cell recorded in TPE confocal (a) and TPE 4Pi type A mode (b,c). The corresponding Fourier transform along the optical axis is also shown (d–f). The fivefold increase in axial resolution ((a) versus (b)) and the correspondingly extended OTF ((d) versus (e)) are immediately visible. The side lobes are well below \({\mathrm{50}}\%\), and the factorization of the PSF's axial and lateral dependence is possible in 4Pi microscopy. An inverse discrete filter can be found, and its application (c) yields a valid and almost artifact-free image (b). Alternatively, lobe removal can be performed in the frequency domain. The Fourier transforms of the raw data (f) are given by the product of the Fourier transform of the lobe-free image (e) and the lobe function \(\hat{l}(k_{z})\). Thus, Fourier transforming the raw data, multiplying by the inverse of the lobe function's Fourier transform, \(\hat{l}^{-1}(k_{z})\;,\)and Fourier back-transforming leads to virtually the same lobe-free image

While an in-depth analysis of nonlinear iterative deconvolution (i. e., image restoration) algorithms is far beyond the scope of this chapter, it can be safely stated that these suffer from similar limitations as the linear deconvolution approaches. They cannot restore information that has been lost in the imaging process. A connected and largely convex OTF that does not imply regions of weak transmission is almost equally important to these algorithms. Nonlinear iterative algorithms take advantage of a priori information about the object that linear deconvolution cannot uncover unambiguously. The simplest a priori information is the positivity of the object and of the image. Carefully applied to 4Pi and I5M data, nonlinear restoration can yield impressive results, but extreme care must be taken not to compromise the reliability of the outcome by false or even biased assumptions.

22.2.3 Improved Axial Resolution in Practice

While the use of coherent beams from two opposing lenses extends the OTF in the axial direction, a detailed analysis shows that there are conditions that must be met to exploit this advantage in an effective manner. Reliable and artifact-free restoration is only possible if the region of the OTF is convex, that is, if it does not contain non-negligible weak regions or gaps. It was shown that the OTF of an SWM exhibits marked gaps along the optical axis that make the removal of the interference artifacts virtually impossible. It is therefore not surprising that previous experimental studies confirmed that in the SWM, it is impossible to unambiguously distinguish two axially separated objects unless the object is thinner than \({\mathrm{50}}\%\) of the wavelength [22.34]. Unambiguous resolution of axially extended objects is impossible with SWM, for fundamental physical reasons. Nevertheless, both SWM and 4Pi microscopy have been successfully applied to the ultra-precise measurement of axial distances [22.58, 22.59, 22.60] and object sizes [22.18, 22.61], which is conceptually and practically less demanding.

The OTF of the I5M is superior to that of the SWM, because it remains nonzero throughout its support. Still, it is weak over a considerable region when compared to the giant zero-frequency peak, which actually is a singularity. Hence, to benefit from this contiguity, I5M data must be recorded with a very large signal-to-noise-ratio. Additionally, the method may be applicable only to sparse non-extended objects, such as points or sparse fine lines. The OTFs of the reported 4Pi microscopes are truly contiguous. In the critical regions, the 4Pi-confocal OTF (Fig. 22.4) exhibits significant values in the \(19{-}32\%\) range. This feature is of key relevance to the removal of the interference artifacts, and hence for unambiguous, object-independent 3-D-microscopy with improved axial resolution.

Another major difference between the effective PSF of the SWM and I5M on the one hand, and the 4Pi-confocal on the other, is the fact that the PSF is spatially much more confined in the latter case. This allows us to reduce the process of lobe removal from a 3-D deconvolution to a one-dimensional linear problem, with the highly useful ancillary effect that single layers can be acquired and deconvolved while all other methods require acquisition of a full 3-D data stack.

The 4Pi concept is the result of rigorously maximizing the aperture angle employed in the imaging process. This results in a superior excitation OTF due to its lateral filling, which is rooted in the fact that the (exciting and detected) light is focused. Hence, while scanning with a focused beam inevitably reduces imaging speed, this procedure also results in fundamentally improved imaging properties of the microscope. Flat field standing wave excitation inherent in the I5M and SWM trades off collected spatial frequencies. The loss of optical frequencies is so significant (Figs. 22.3 and 22.4) that the ability to provide unambiguous axial resolution is either put at risk or simply not viable.

Nevertheless, improved axial resolution and imaging similar to the 4Pi microscope is reportedly possible when combining I5M with offline image restoration [22.41]. The combination of I5M with fringe-pattern illumination and subsequent image restoration has also been suggested to (i) alleviate the problem of the zero-frequency singularity [22.44, 22.62] and (ii) increase the lateral resolution to that of (restored) confocal microscopy. Leaving aside the technical complexity of controlling the interference patterns of typically four pairs of beams, this suggestion confirms that an unambiguous axial resolution requires the employment of a wider angular spectrum. In fact, the spherical beams in a 4Pi microscope can be regarded as a complete spectrum of interfering plane waves coming from all angles available. Hence, from the standpoint of imaging theory, the combination of I5M with fringe-pattern illumination is a modification of I5M towards a scheme that is more similar to the 4Pi arrangement; the improvements in the OTF are gained by conditions that converge to the focusing conditions found in the 4Pi microscope.

In real samples, the OTF of the microscopes compromises aberrations that were not included in our comparison of the concepts. Residual misalignments of the foci induced by variations in the refractive index in the sample play a role. However, successful 4Pi imaging of the mouse fibroblast cytoskeleton [22.48] and the I5M imaging of similar structures [22.41] have revealed that aberration effects are surmountable. Image deconvolution requires prior knowledge of the PSF and hence its explicit determination with a point-like object. This is particularly important, since the PSF depends on the relative phase of the two counter-propagating wave fronts. It has been shown that in 4Pi microscopy, the type of interference in the focal plane (constructive, destructive, or anything in between) is of lesser importance [22.49]. Depending on its influence on the OTF, the same will also apply to I5M. Thus far, the structure of the PSF has been determined by measuring the response of test objects such as fibers or fluorescent beads. However, it has been shown that the relative phase can be extracted directly from the image data [22.63]. This is of great importance in cases where the imaged object itself alters the relative phase of the interfering beams.

A filled OTF entails robustness in operation and lower amenability to potential misalignment. Thus, 4Pi microscopy enabled the application of interference microscopy to super-resolved imaging of specimens in aqueous media using water immersion lenses [22.43]. This is noteworthy, since water lenses feature semi-aperture angles of not more than \(64^{\circ}\), as compared with \(68^{\circ}\) available for oil or glycerol immersion, therefore increasing side lobes and rendering the problem of missing frequencies in the OTF more severe. Imaging a watery sample is one of the prerequisites of live-cell imaging.

For realistic aperture angles, the inherent lack of contiguity of the OTF in the SWM renders the removal of interference ambiguities impossible. The I5M becomes more viable through filling the frequency gaps present in the SWM, albeit with values that are weak with respect to the zero-frequency components. Both systems have yet to prove their applicability in live-cell imaging. If lenses of \(70^{\circ}\) and higher were available, the robustness and applicability of I5M would increase significantly. This method would then become viable for a much larger variety of objects, at least for those that were not too dense or too 3-D-convoluted. The fact that the enlargement of the semi-aperture angle is the key to the performance of the I5M highlights the exploitation of the entire spherical wave fronts as the central physical element. In a sense, the I5M can be viewed as a 4Pi system that maximizes the degree of parallelization in the focal plane.

Parallelization is readily accomplished in 4Pi and 4Pi-confocal microscopy as well. A multifocal variant of TPE 4Pi-confocal microscopy, namely multifocal multiphoton 4Pi microscopy [22.18] ( ), has indeed translated the typical \({\mathrm{140}}\,{\mathrm{nm}}\) axial resolution of a TPE 4Pi microscope into live-cell imaging. Importantly, although present and noticeable, phase changes induced by the cell proved more benign than anticipated. Nevertheless, phase alterations and wave-front aberrations due to refractive index changes within the specimen are likely to confine 4Pi microscopy and related techniques to the imaging of individual cells or thin cell layers. In conjunction with nonlinear image restoration, this imaging method has displayed \({\mathrm{100}}\,{\mathrm{nm}}\) 3-D-resolution under live-cell conditions. For example, MMM-4Pi microscopy has provided superior 3-D images of the reticular network of green fluorescent protein ( )-labeled mitochondria in live budding yeast cells.

4Pi microscopy has been realized in a compact optical unit that was interlaced with a state-of-the-art single-beam scanning confocal fluorescence microscope (Leica TCS SP2, Mannheim, Germany). Operating in the type C mode, i. e., coherent spherical wave fronts both for excitation and for confocal detection, this compact and rugged system displayed sevenfold improved axial resolution (\({\mathrm{80}}\,{\mathrm{nm}}\)) over confocal microscopy in live cells [22.64]. This system became commercially available as the first far-field optical microscope with axial super-resolution.

The fact that single-photon excitation 4Pi-confocal microscopy has not been pursued fully in further experimental developments can be attributed to the superiority of the OTF in the TPE version. Nevertheless, it is clear from imaging theory that a single-photon 4Pi-confocal microscope of type C features a more contiguous and better-filled OTF than an I5M, which inevitably operates in the single-photon mode. Given further improvements in aberration compensation and lens manufacturing, it is feasible that single-photon excitation 4Pi microscopy (of type C) could also be realized in a reliable manner. An inherent advantage of single-photon excitation over its two-photon counterpart is that the much shorter excitation wavelengths lead to axially narrower focal spots. For example, using a wavelength of \(\approx{}{\mathrm{400}}\,{\mathrm{nm}}\), a single-photon 4Pi microscope of type C would deliver an axial resolution of \(\approx{}{\mathrm{60}}\,{\mathrm{nm}}\). This number further underscores the potential for 4Pi microscopy to deliver 3-D images with axial sections well in the tens-of-nanometers regime.

In summary, the resolution of a far-field fluorescence light microscope can be improved down to the range of \(60{-}100\,{\mathrm{nm}}\) along the optical axis by coherently adding the focal light fields of two opposing lenses. The addition of the fields leads to improved resolution for both excitation at the focal point and detection of fluorescence at a common point. The axial resolution is improved only if the main maximum of the resulting PSF is at least twice as large as the primary side maxima arising from the coherent addition. The latter condition can only be fulfilled if, at least for one of the processes, the added light field is a spherical wave front covering the aperture angle of the lens. This in turn shows that the key physical element for improving the axial resolution by the coherent use of two opposing lenses is not the production of an interference pattern, but the enlargement of the aperture of the system.

22.3 Breaking the Diffraction Barrier:STED Microscopy and the RESOLFT Concept

Increasing the total aperture of the focusing system with two opposing lenses improves the 3-D resolution of a far-field microscope, but does not break the diffraction barrier. In contrast, 4Pi microscopy exploits the full potential of diffraction-limited imaging. This particularly applies to (multiphoton) type C 4Pi-confocal microscopy, which can be regarded as the far-field optical microscope with the largest possible aperture. For a microscope, being subject to the diffraction limit implies that the system features the maximum resolution that cannot be exceeded. On the other hand, breaking the diffraction barrier implies the potential for molecular resolution, or an OTF approaching an infinitely large bandwidth.

The first concept to break the diffraction barrier was stimulated emission depletion (STED) fluorescence microscopy. The physical concept of STED microscopy was described as early as 1994 [22.9] and was soon followed by ground state depletion (GSD) microscopy, a related concept that also entailed diffraction-unlimited resolution [22.10]. Whereas STED microscopy utilized stimulated emission to deplete the excited state of the fluorophore, GSD aimed at depleting its ground state by transiently pumping the dye into a long-lived dark state, e. g., its triplet state. Importantly, both concepts share the same principle for breaking Abbe's barrier: a focal intensity distribution featuring a zero point (or at least a strong gradient in space) effects a saturated depletion of one of the molecular states that are essential to the fluorescence [22.10, 22.11]. Following depletion, the state is populated again, that is, the state transition is reversible.

An even closer examination of the underlying concept shows that in fact any saturable transition between two states where the molecule can be returned to its initial state is a potential candidate for breaking the diffraction barrier [22.11, 22.12, 22.65]. The general concept has therefore been termed reversible saturable/switchable optically linear (fluorescence) transition (RESOLFT). The transition utilized can be selected to match the practical conditions of the imaging problem at hand, such as the required intensities, the available light sources, and the avoidance of photobleaching. An important aspect with respect to biological applications is the compatibility with living cells.

The basic idea of the RESOLFT concept can be understood by considering a molecule with two arbitrary states A and B between which the molecule can be transferred. In fluorophores, typical examples for these states are the ground and first excited electronic states, and conformational and isomeric states. The transition A\({\rightarrow}\)B is induced by light, but no restriction is imposed regarding the transition B\({\rightarrow}\)A. It may be spontaneous, but may also be induced by light, heat, or any other mechanism. The only further assumption is that at least one of the two states is critical to the generation of the signal. In fluorescence microscopy, this means that the dye can fluoresce only (or much more intensely) in state A. The way in which such a system may be exploited to generate diffraction-unlimited resolution in fluorescence imaging (or any kind of manipulation, probing, etc., that depends on one of the states) is illustrated in Fig. 22.6a-d.

Fig. 22.6a-d

The RESOLFT principle. Diffraction-unlimited spatial resolution is achieved by saturating a linear but reversible optical transition from state A to state B. The simple explanation: A uniform dye distribution of state A is illuminated by a strongly modulated intensity light distribution that transfers the dye molecules to state B; ideally the modulation is perfect, so that the local intensity minima are actually zeros. Here, we choose the narrowest possible modulation \(I(r)=\cos^{2}(2\uppi r/\lambda)\). The left-hand panels show \(I(r)\) and the resulting probability \(N_{\mathrm{A}}(r)\) of the dye molecules being in state A. The right-hand panels depict the dependence of \(N_{\mathrm{A}}(r)\) on the local intensity \(I(r)\), exhibiting the typical saturation behavior. \(I_{\mathrm{S}}\) is defined as the intensity where half of the molecules are transferred to state B. (a) At \(I(r)<I_{\mathrm{S}}\), the modulation of the light is replicated in \(N_{\mathrm{A}}(r)\). The narrowest distribution of \(N_{\mathrm{A}}(r)\) will also be limited by diffraction, because \(I(r)\) is diffraction-limited and the relationship between \(I\) and \(N_{\mathrm{A}}\) is basically linear. (b) Increasing the maximum intensity moves the points where \(I(r)\) reaches \(I_{\mathrm{S}}\) closer to the local intensity minimum, e. g., the zero. Because these are also the points where \(N_{\mathrm{A}}(r)\) drops to 0.5, the FWHM of \(N_{\mathrm{A}}(r)\) is correspondingly reduced. (c) Further increasing the maximum intensity beyond \(I_{\mathrm{S}}\) leads to a further reduction of the FWHM. If A is the fluorescent state, \(N_{\mathrm{A}}\)(r) is read out as the desired signal stemming from a narrow region. In order to obtain an image, the local minima are scanned across the sample. If the signal stems from state A, the intensity values \(N_{\mathrm{A}}(r)\) are simply read out subsequently and the image is assembled in a computer. If the signal is generated by the majority population in state B (e. g., state B is the fluorescent state), the function \(1-N_{\mathrm{A}}(r)\) is read out and the image must be inverted later. This approach is challenged by signal-to-noise issues. (d) There is no theoretical limit to this method, and its resolution is ultimately determined by the available laser power and the potentially limiting photodamage

We begin with all molecules or entities in the sample being in state A. Our goal is to generate a diffraction-unlimited distribution of molecules in state A. To this end, the sample is illuminated with light that drives the transition from A to B. The intensity distribution \(I(r)\) of the illuminating light is of course diffraction-limited but features one or several positions with zero intensity. After illumination, the probability \(N_{\mathrm{A}}(r)\) of finding molecules in state A depends on \(I(r)\) and displays peaks where \(I(r)\) is zero and no transitions to B are induced. If we choose the maximum intensity \(I_{\mathrm{max}}\) in such a way that it is many times (\({\zeta}\) times, \({\zeta}=I_{\mathrm{max}}/I_{\mathrm{S}}\) is called the saturation factor) higher than the threshold intensity \(I_{\mathrm{S}}\) of the transition A\({\rightarrow}\)B (the threshold at which \({\mathrm{50}}\%\) of the molecules are transferred into state B), then molecules will be almost exclusively in state B even at positions where \(I(r)\) is only a small fraction (\(\approx{}5/{\zeta}\)) of \(I_{\mathrm{max}}\). This means that while molecules remain in state A in the intensity zeros, they are transferred to state B even at locations in the immediate vicinity of the zero. Thus, the peaks of \(N_{\mathrm{A}}(r)\) become very narrow, featuring steep edges at the same time. Figure 22.6a-d is presented in one dimension for clarity, but this idea is readily extended to all directions in space, and hence to 3-D imaging.

The Fourier transform (the effective excitation OTF) of an increasingly sharp peak correspondingly becomes wider with an increasing \(I_{\mathrm{max}}\). A complementary mathematical explanation for this fact is that higher-order contributions that are described by several auto-convolutions of the depletion pattern become increasingly important for higher intensities. This is in contrast to very low intensities, where the dependence of \(N_{\mathrm{A}}\) on the depletion intensity is approximately linear: \(N_{\mathrm{A}}(r)\cong{}1-I(r)/I_{\mathrm{S}}\). Here, a single additional convolution is introduced in frequency space, and the theoretical cutoff of the support can only be pushed from the \(4k\) we derived for conventional imaging to \(6k\). Thus, the diffraction barrier is not truly broken for low intensities, but only shifted to a slightly higher value.

The generated probability distribution can be used for high-resolution imaging by applying scanning. Since the fluorescence stems exclusively from the immediate vicinity of the positions where \(I(r)\) is zero, scanning the zero(s) through the sample with simultaneous recording of the fluorescence signal allows one to assemble an image with basically unlimited spatial resolution in the far-field. For particular conditions, the resolution of this process is determined by the FWHM of the peaks in \(N_{\mathrm{A}}(r)\), which in turn is determined solely by the saturation factor. The FWHM can become infinitely narrow, that is, down to the size of a single molecule. The reason that Abbe's spatial frequency cutoff does not apply in this case is simply the fact that in the RESOLFT concept, the lens acts merely as a condenser collecting the fluorescence. For example, when scanning with a single zero, one can also imagine placing the detector right next to the sample in order to measure \(N_{\mathrm{A}}(r)\). Scanning in the sense of a sequential readout is absolutely mandatory, because—and Abbe was perfectly correct in this regard—the objective lens cannot transmit the higher spatial frequencies through the lens.

However, this does not imply that a RESOLFT microscope needs to be based on single-beam scanning. Parallelization is readily possible, and hence detection with a conventional camera is feasible if the nodes are farther apart than the classical resolution limit of the microscope. When the sharply localized fluorescence from the nodes is imaged onto the camera, the resulting spot will certainly be blurred as a result of diffraction, and will possibly extend over several pixels on the camera. However, if the nodes are farther apart than Abbe's diffraction barrier, each of the blurred spots can be assigned to the respective nodes. Sequential read-out of the camera allows one to integrate the measured signal of each blurred spot or line separately and to associate it with the corresponding sharply localized region in the sample from which the signal emerged.

Importantly, implementing such a wide-field detection in a microscope does not imply that sub-diffraction resolution is possible in conventional microscopy, since scanning is still an essential element in the process of image formation. In a sense, wide-field is not the perfectly appropriate term in any event, since RESOLFT-based concepts always imply that a part of the sample (at the very least, one single point) is omitted from the saturable transition. Consequently, camera-based RESOLFT approaches are essentially parallelized scanning concepts.

This scanning wide-field detection is readily explained in frequency space as well. The PSF describing the generated probability of the molecules of being in state A, \(N_{\mathrm{A}}(r)\), consists of periodic sharp peaks, and can be described as a convolution of a single peak with a comb function or with its multidimensional equivalent. In frequency space, this is equivalent to a multiplication of the broad frequency band of a single peak with a comb function. Thus the Fourier transform of \(N_{\mathrm{A}}(r)\) which dominates the excitation OTF is given by delta peaks separated by the inverse distance of the nodes in \(I(r)\). The speed with which they drop off towards large frequencies depends on the width of a single peak in \(N_{\mathrm{A}}(r)\). The effective OTF is given by the convolution of this series of delta peaks with the detection OTF. Therefore, it will be continuous whenever the usable support of the latter is larger than the distance between the delta peaks in the excitation OTF. Not surprisingly, this simply means that the intensity zeros of \(I(r)\) are far enough apart that the detection system can separate their fluorescence.

Please note that complete depletion of A (or complete darkness of B) is not required. It is sufficient that the non-nodal region features a constant, notably lower probability of emitting fluorescence, so that it can be distinguished from its sharp counterpart. Even if B, and not A, is the brighter state, one can read out B and may obtain the same super-resolved image after mathematical post-processing that entails subtraction of signals from each other. Although reading out B, in principle, also delivers unlimited resolution, this version is heavily challenged by signal-to-noise issues. This stems from the fact that the bright light from the non-nodal regions contributes with a substantial amount of photon shot noise.

It is also important to keep in mind that the nonlinearities introduced in these concepts are not analogous to the nonlinear interactions, for example, connected with \(m\)-photon excitation, \(m\)-th harmonics generation, or coherent anti-Stokes-Raman scattering [22.2, 22.66]. In the latter cases, the nonlinear signal stems from the simultaneous action of more than one photon at the sample, which would only work at high focal intensities. In contrast, the nonlinearity brought about by saturation and depletion arises from a change in the population of the involved states, which is effected by a single-photon process, namely stimulated emission. Therefore, unlike in \(m\)-photon processes, strong reductions in the region containing molecules in state A are achieved at comparatively low intensities.

Next, let us derive an estimate for the resolution achievable in such a system at finite depletion intensities [22.13, 22.67]. We denote the rates of A\({\rightarrow}\)B and B\({\rightarrow}\)A with \(k_{\mathrm{AB}}\) and \(k_{\mathrm{BA}}\), respectively. The time evolution of the normalized populations of the two states \(n_{\mathrm{A}}\) and \(n_{\mathrm{B}}\) is then given by
$$\frac{\mathrm{d}n_{\mathrm{A}}}{\mathrm{d}t}=-k_{\mathrm{AB}}n_{\mathrm{A}}+k_{\mathrm{BA}}n_{\mathrm{B}}=\frac{-\mathrm{d}n_{\mathrm{B}}}{\mathrm{d}t}\;.$$
(22.20)
Independently from its initial state, after a time
$$t\geq\left(k_{\mathrm{BA}}+k_{\mathrm{AB}}\right)^{-1}\;,$$
(22.21)
the equilibrium is approximately reached and the population of state A is given by
$$N_{\mathrm{A}}=\frac{k_{\mathrm{BA}}}{k_{\mathrm{BA}}+k_{\mathrm{AB}}}\;.$$
(22.22)
Now the process described in Fig. 22.6a-d begins. State A is depleted at a rate \(k_{\mathrm{AB}}={\sigma}I\), where \({\sigma}\) denotes the molecular cross section, and the intensity \(I\) is written as photon flux per unit area. Hence, the equilibrium population is given by
$$N_{\mathrm{A}}(r)=\frac{k_{\mathrm{BA}}}{\sigma I(r)+k_{\mathrm{BA}}}\;.$$
(22.23)
And \(N_{\mathrm{A}}=1/2\) for the threshold intensity
$$I_{\mathrm{S}}=\frac{k_{\mathrm{BA}}}{\sigma}\;.$$
(22.24)
From (22.23) we see that where \(I(r)\gg{}I_{\mathrm{S}}\), all molecules end up in B. Thus, if we choose
$$I(r)=I_{\max}f(r)\;,$$
(22.25)
with \(I_{\mathrm{max}}\gg{}I_{\mathrm{S}}\), molecules in state A are only found in the nodes of the diffraction-limited distribution function \(f(r)\). As an example, we choose a sine-square intensity distribution such as produced by a standing wave
$$f(x)=\sin^{2}\left(2\uppi n\frac{x}{\lambda}\right)$$
(22.26)
for illumination, where \(n\) denotes the index of refraction of the medium. A simple calculation shows that the FWHM of the peaks of \(N_{\mathrm{A}}\), and hence the resolution of the microscope, is then given by
$$\Updelta x=\frac{\lambda}{\uppi n}\arcsin\left(\frac{1}{\zeta}\right)\cong\frac{\lambda}{\uppi n\sqrt{\zeta}}\;.$$
(22.27a)

A saturation factor of \({\varsigma}=1000\) yields \({\Updelta}x\approx{}{\lambda}/100\), but in principle the spot of A molecules can be continuously squeezed by increasing \(\varsigma\). Scanning with such a spot and simultaneous recording of the signal from it delivers diffraction-unlimited resolution. Equation (22.27a) quantitatively describes this possibility in a microscope using diffraction-limited beams.

If the intensity distribution \(I(r)\) is produced by a lens, the largest frequency will be determined by the numerical aperture of the lens \(n\sin\alpha\), due to diffraction. In this case, (22.27a) changes into
$$\Updelta x\cong\frac{\lambda}{\uppi{}n\sin\alpha{}\sqrt{\zeta}}\;.$$
(22.27b)
If at the same time the molecules are being driven back from B to A by a light intensity distribution following a cosine-square form, we have
$$k_{\mathrm{BA}}=\sigma_{\mathrm{BA}}I^{\mathrm{BA}}_{\max}\cos^{2}\left(2\uppi n\sin(\alpha)\frac{x}{\lambda}\right),$$
with \(\sigma_{\mathrm{BA}}\) denoting the cross section of the transition. By defining the threshold intensity as
$$I_{\mathrm{S}}=\frac{\sigma_{\mathrm{BA}}I_{\mathrm{BA}}^{\max}}{\sigma},$$
we obtain
$$\Updelta x\cong\frac{\lambda}{\uppi n\sin\alpha\sqrt{1+\zeta}}\;.$$
(22.27c)

This equation is particularly appealing, since for a vanishing saturation factor \(\zeta=0\) it virtually assumes Ernst Abbe's diffraction-limited form, whereas for \(\zeta\rightarrow\infty\) the spot becomes infinitely small.

There are also adverse effects that need to be taken into account. The first is that state A cannot be completely emptied even by very intense illumination (e. g., because there is an excitation by the same beam), and the second is that state B may also contribute to the signal. Both can be considered by including a constant offset in (22.23)
$$N_{\mathrm{A}}(r)=\frac{(1-\delta)k_{\mathrm{BA}}}{\sigma I(r)+k_{\mathrm{BA}}}+\delta\;.$$
(22.28)
This would result in the image consisting of a super-resolved image plus a (weak) conventional image. The latter does not alter the frequency content of the image. Therefore, given sufficient SNR, the resolution will not deteriorate. In other words, if \({\delta}\) is sufficiently small so as not to swamp the image with noise, the conventional contribution can be subtracted [22.10, 22.11].
A further experimental problem is caused by imperfections in the intensity zeros. Imagine the standing wave is aberrated and approximately described by
$$f(x)=(1-\gamma)\sin^{2}\left(2\uppi n\sin(\alpha)\frac{x}{\lambda}\right)+\gamma\;.$$
(22.29)
Such zeros with insufficient depth turn out to have a more serious impact on performance. The maximum signal in the intensity minima drops by a factor \((1+\zeta{}\gamma)\) as a result. Following the same calculation as above, we obtain
$$\Updelta x=\frac{\lambda}{\uppi n\sin(\alpha)}\sqrt{\gamma+\frac{1}{\zeta}}\;,$$
(22.30)
and therefore the maximum achievable resolution is given by \({\lambda\sqrt{\gamma}}/({\uppi}n\sin(\alpha))\). Even at \(\gamma\approx{}{\mathrm{1}}\%\) resolutions of \({\lambda}/20\) at saturation factors of \(\mathrm{100}\) can be achieved without losing more than half the signal in the intensity minima. In practice, smaller \(\gamma\) can usually be achieved.
While RESOLFT is far more intuitively explained in the way presented above, it is also helpful to take a look at the frequency space in order to relate these insights to the concepts and results presented in the first part of this chapter. The dependence of the effective excitation PSF, i. e., the distribution of the probability that a molecule actually emits a fluorescence photon, is governed by the saturation-level-dependent value of \(N_{\mathrm{A}}(r)\) expressed in (22.23). If for \(I(r)=0\) the microscope begins, for example, with a conventional PSF \(h_{\mathrm{c}}(r)\) used for imaging the distribution \(N_{\mathrm{A}}(r)\) onto a camera, the effective PSF of the system is given by
$$h(r)=h_{\mathrm{c}}(r)\frac{1}{{1+\sigma I(r)}/{k_{\mathrm{BA}}}}\;.$$
(22.31)
Now let \({\sigma}I_{\mathrm{max}}/k_{\mathrm{BA}}=\zeta\) and \(I(r)=I_{\max}f(r)\) as above; then we can expand (22.31) in a Taylor series
$$h(r)=h_{\mathrm{c}}(r)\sum_{\nu}\xi^{\nu}g(r)^{\nu}\frac{1}{\zeta+1}\;,$$
(22.32)
where \(g(r)=1-f(r)\) and \({\xi}={\zeta}/({\zeta}+1)\), and we obtain the OTF after Fourier transformation
$$\begin{aligned}\displaystyle o(k)=\smash[b]{\frac{\hat{h}_{\mathrm{c}}(k)}{1+\zeta}}\otimes(&\displaystyle\delta(k)+\xi\hat{g}+\xi^{2}\hat{g}\otimes\hat{g}\\ \displaystyle&\displaystyle+\xi^{3}\hat{g}\otimes\hat{g}\otimes\hat{g}+\ldots{}).\end{aligned}$$
(22.33)
At low intensities, \(\zeta\), and therefore \({\xi}\), is so small that only the linear term is relevant, and the convolution extends the support to \(6k\), as discussed above. The larger the maximum intensity, the more important higher orders of the Taylor series will become. These involve multiple auto-convolutions of the function \(g\) extending the support further and further.
While a quantitative treatment in frequency space is more complicated and less intuitive than the one introduced in the previous section, the following analysis gives a feel for the effect of the saturation factor and also illustrates the possible vast expansion of the OTF support. For the sake of simplicity, we assume a Gaussian form of the light distribution function
$$f(x)=1-\exp\left(\frac{-x^{2}}{2a^{2}}\right).$$
(22.34)
The properly normalized \(m\)-fold auto-convolution of \(g\) is then given by
$$\otimes_{\mathrm{m}}\hat{g}(k)=a\sqrt{\frac{2\uppi}{m}}\exp\left(\frac{-a^{2}k^{2}}{2m}\right).$$
(22.35)
Now let us assume that the useful support ends at a frequency where the OTF is attenuated to a small fraction \({\varepsilon}\) of its value at small frequencies. For large saturation factors, the influence of the convolution with the confocal OTF on the cutoff frequency can be neglected, and we have to calculate the sum in brackets in (22.33). Substituting (22.35) into (22.33) and approximating the sum by an integral, for the term in brackets we get
$$o(k,\zeta)\cong-\mathrm{i}a\sqrt{\frac{1}{2\ln\xi(1+\zeta)}}\exp\big({-}a|k|\sqrt{2\ln\xi}\big)\;.$$
(22.36)
For large saturation factors, we can write
$$\ln\xi=\ln\zeta-\ln(\zeta+1)\cong-\frac{1}{\zeta}$$
and obtain
$$o(k,\zeta)\cong\frac{a}{\sqrt{2\zeta}}\exp\left(-a|k|\sqrt{\frac{2}{\zeta}}\right).$$
(22.37)
This means that the attenuation of the modulus of the OTF at large frequencies is inversely proportional to the square root of the saturation factor. This is equivalent to saying that the resolution increases with the square root of the saturation factor just as we expected from our previous analysis, the situation explained in Fig. 22.6a-d.

22.3.1 STED Microscopy: The First Diffraction-Unlimited Concept

STED microscopy produces sub-diffraction resolution and sub-diffraction-sized fluorescence volumes in exactly the manner described above by the depletion of the fluorescent state of the dye. Depletion inherently implies saturation of the depleting transition. In most cases, STED microscopy has been realized in a (partially confocalized) spot-scanning system due to a number of technical advantages, but it has been conceptually clear from the outset that non-confocalized detection is viable as well [22.9]. The principle, a schematic setup, and an example of a measurement of the resolution increase are shown in Fig. 22.7a-e. The fluorophore in the fluorescent state \(\mathrm{S}_{1}\) (state A) is stimulated to the ground state \(\mathrm{S}_{0}\) (state B) with a doughnut-shaped beam. The saturated depletion of \(\mathrm{S}_{1}\) confines fluorescence to the central intensity zero. With typical saturation intensities in the range of \(1{-}100\,{\mathrm{MW/cm^{2}}}\), saturation factors of up to 120 have been reported [22.68, 22.69]. This should yield a 10-fold resolution improvement over the diffraction barrier, but imperfections in the doughnut limited the improvement to \(5{-}7\)-fold in initial experiments [22.69].

Fig. 22.7a-e

Stimulated emission depletion (STED) microscopy was the first implementation of the RESOLFT principle. (a) Dye molecules are excited to state A by an excitation laser pulse. (b) Fluorescence is detected over most of the emission spectrum. However, molecules can be quenched back into the ground state B using stimulated emission before they fluoresce by irradiating them with a light pulse at the edge of the emission spectrum shortly after the excitation pulse and before they are able to emit a fluorescence photon. Saturation is realized by increasing the intensity of the depletion pulse and consequently inhibiting fluorescence everywhere except at the zero points of the focal distribution of the depletion light. (c) Schematic of a point-scanning STED microscope. Excitation and depletion beams are combined using appropriate dichroic mirrors (DC s). The excitation beam forms a diffraction-limited excitation spot in the sample (inset in (d)), while the depletion beam is manipulated using a phase plate (PP) or any other device to tailor the wave front (left inset in (e)). The resulting quenching probability when saturating the depletion process. In the right inset in (d) and the right inset in (e), experimental comparison between the confocal PSF and the effective PSF after switching on the depleting beam is shown. Note the doubled lateral and fivefold-improved axial resolution. The reduction in dimensions (\(x,y,z\)) yields ultrasmall volumes of sub-diffraction size, here \({\mathrm{0.67}}\,{\mathrm{aL}}\) [22.68], corresponding to an 18-fold reduction compared to its confocal counterpart. The spot size is not limited on principle grounds but by practical circumstances, such as the quality of the zero and the saturation factor of depletion

As already stated, light microscopy resolution can be described either in real space or in terms of spatial frequencies. In real space, the resolution is assessed by the FWHM of the focal spot; more precisely, the extent of the region, where the fluorescent state A is assumed by the fluorophores. The measurements depicted in Fig. 22.8 were carried out with an excitation wavelength of \({\lambda}={\mathrm{635}}\,{\mathrm{nm}}\), an oil immersion lens with a numerical aperture of \(\mathrm{1.4}\), and with the smallest possible probe: a single fluorescent molecule [22.14, 22.28]. Figure 22.8a shows the measured profile of the PSF in the focal plane (\(x\)) for a conventional fluorescence microscope, along with its sharper sub-diffraction STED fluorescence counterpart. STED leads to an improvement in resolution by a factor of approximately \(\mathrm{5.5}\).

Fig. 22.8

(a) Comparison of the effective PSF's lateral intensity profile for confocal and STED microscopy indicating an \(\approx{}5.5\)-fold resolution increase in the latter. (b) Lateral cuts through the effective OTFs giving the bandwidth of the lateral spatial frequencies passed to the image. The data plotted in (a) and (b) are gained by probing the fluorescent spot of a scanning microscope with a single molecule of the fluorophore JA 26 using a numerical aperture of \(\mathrm{1.4}\) (oil) objective lens and at wavelengths of \({\mathrm{635}}\,{\mathrm{nm}}\) (excitation), \(650{-}720\,{\mathrm{nm}}\) (fluorescence collection), and \({\mathrm{790}}\,{\mathrm{nm}}\) (STED). The inlay demonstrates sub-diffraction resolution with STED microscopy. Two identical molecules located in the focal plane that are only \({\mathrm{62}}\,{\mathrm{nm}}\) apart can be entirely separated by their intensity profile in the image. A similarly clear separation by conventional microscopy would require the molecules to be at least \({\mathrm{300}}\,{\mathrm{nm}}\) apart. Data adapted from [22.28]

Figure 22.8b shows the OTF of a conventional microscope along with the enlarged OTF of the STED fluorescence microscope. As expected, the effective OTF's support in the confocal case ends at approximately
$$({\mathrm{2/635}}\,{\mathrm{nm}}+{\mathrm{2/720}}\,{\mathrm{nm}})=5.91/{\mathrm{\upmu{}m}}.$$
For the case of STED, we included the OTF after successful linear deconvolution, which restores higher spatial frequencies that are not swamped by noise. The region of usable OTF support is approximately marked by the region where frequencies are enhanced by the deconvolution process without producing artifacts and is \(\approx{}5.5\) times as large as that for the confocal case. This marks a fundamental breaking of Abbe's diffraction barrier in the focal plane. The inlay demonstrates the resulting sub-diffraction resolution exemplified by the linearly deconvolved STED image of two molecules at a distance of \({\mathrm{62}}\,{\mathrm{nm}}\). They are distinguished in full by two narrow peaks [22.28]. As a result of deconvolution, the individual peaks are sharper (\({\mathrm{33}}\,{\mathrm{nm}}\) FWHM) than the initial peak of \({\mathrm{40}}\,{\mathrm{nm}}\) FWHM.

Utilizing STED light wavelengths of \(\lambda=750\)\({\mathrm{800}}\,{\mathrm{nm}}\), a lateral resolution of down to \({\mathrm{16}}\,{\mathrm{nm}}\) was achieved as early as 2005 in experiments with single JA 26 molecules spin-coated on a glass slide [22.14]. Measuring the resolution as a function of the applied STED intensity confirmed the predicted increase of the resolving power with the square root of the saturation factor (Fig. 22.9a-c). Of course, the cutoffs presented are based on a somewhat arbitrary definition of what can be considered usable frequencies at a certain signal-to-noise ratio. However, regardless of how low this threshold is set, the confocal support cannot extend beyond \(\approx{}6/{\mathrm{\upmu{}m}}\), while the STED OTFs' support is theoretically unlimited.

Fig. 22.9a-c

STED microscopy reduces the region from which fluorescence can originate to a degree far below the diffraction limit: (a) Spot of a confocal microscope compared with that in a STED microscope utilizing a \(y\)-oriented intensity valley for STED squeezing the spot in the \(x\)-direction to \({\mathrm{16}}\,{\mathrm{nm}}\) width. (b) Bandwidth in STED is fundamentally increased over confocal microscopy. For the 1-D depletion scheme, the usable support of the OTF is increased by almost a factor of \(\mathrm{8}\). When using a doughnut-shaped depletion beam with a wider-intensity zero, the OTF support is still extended almost fivefold. (c) The average fluorescent spot size \(\delta\) decreases with the STED intensity following a square root law, in agreement with (22.27a). Because the resolution depends on molecule orientation, the spot sizes were measured for several tens of single molecules. The curve follows the mean values (squares) and the inset shows the histogram of the measured spot sizes at \({\mathrm{1100}}\,{\mathrm{MW/cm^{2}}}\), with the minimal FWHM at \({\mathrm{16}}\,{\mathrm{nm}}\) and a \({\mathrm{26}}\,{\mathrm{nm}}\) average

The one-dimensional ( ) phase plate yielding \({\mathrm{16}}\,{\mathrm{nm}}\) resolution is optimized for maximum resolution improvement in the lateral direction perpendicular to the polarization of the depleting light and leads to an intensity distribution with two strong peaks at either side of the excitation maximum [22.70]. Because the depleting light is polarized, the resolution gain depends on the orientation of the molecules. However, a considerable increase in resolution is still possible for the second phase plate, which yields a doughnut-shaped intensity distribution and thus an almost isotropic resolution increase in the lateral directions when using circularly polarized light.

In order to squeeze the fluorescence spot in both lateral directions, two STED beams aberrated with 1-D phase plates oriented at \(90^{\circ}\) to each other can be combined. Together with circularly polarized excitation, almost uniform resolution in the focal plane is achieved, as shown in Fig. 22.10a-i. A series of \(xy\) images acquired with different STED beam powers demonstrate the resolution increase and concomitant widening of the OTF when the applied saturation factor increases. This combination of two incoherent beams causes the resolution to depend on the orientation of the transition dipole, and results in spikes along the \(x\)- and \(y\)-directions of the OTF when imaging randomly oriented fluorophores (Fig. 22.10a-i). Newer phase plates avoid such effects and improve the effective saturation factors at a given total STED power. Incoherent combination can then be used to improve the resolution in all three spatial dimensions. The resulting PSFs exhibit very weak dependence on dipole orientation [22.70] and enable the application of STED to the imaging of biological specimens and reliable subsequent linear deconvolution [22.71].

Fig. 22.10a-i

Images of a wetted \(\mathrm{Al_{2}O_{3}}\) matrix featuring \(z\)-oriented holes with a spin-casting of a dyed (JA 26) polymethyl methacrylate solution. The rings formed in this way are \(\approx{}{\mathrm{250}}\,{\mathrm{nm}}\) in diameter and barely resolved in confocal mode. (a–d) show the confocal image (a) and STED images with two depleting beams perpendicularly polarized and aberrated by 1-D phase plates (b–d). The excitation PSF (g) and the STED PSF for \(y\)-polarization (h) and \(x\)-polarization (i) are also shown. The STED intensity was chosen at the spots marked in the saturation curve (f). The smaller effective spot size also results in an extended OTF. Here, the inlays show the two-dimensional () Fourier transformation of the images on the left and the graphs show a profile along the \(x\)-direction. Note the logarithmic scales. The Fourier transform of the image is given by the product of OTF and the Fourier transform of the object. For such regular structures, an estimate for the modulus of the OTF can therefore be gained by estimating the latter and solving for the OTF. Dashed line shows the Fourier transform of a ring with a diameter of \({\mathrm{275}}\,{\mathrm{nm}}\) and a width of \({\mathrm{50}}\,{\mathrm{nm}}\), and the estimated OTF is presented in (e). (f) Suppression of fluorescence resulting from stimulated emission. The phase plates were removed, and the ratio of fluorescence without STED light (\(F_{0}\)) and with the STED beams switched on (\(F\)) was recorded. The intensities are pulse intensities per beam at the global maximum

The first sub-diffraction images of biological samples, with threefold enhanced axial and doubled lateral resolution, were obtained with membrane-labeled bacteria and live budding yeast cells [22.68]. While there is evidence for increased nonlinear photobleaching of some dyes when increasing the depletion intensity [22.72] the intensities applied are not detrimental to living cells. This is not surprising, since the intensities are two to three orders of magnitude lower than those used in multiphoton microscopy [22.4]. Moreover, STED has proven to be single-molecule-sensitive, despite the proximity of the STED wavelength to the emission peak. In fact, individual molecules have been switched on and off by STED upon command [22.73, 22.74].

The power of STED and 4Pi microscopy was synergistically combined to demonstrate for the first time an axial resolution of \(30{-}40\,{\mathrm{nm}}\) in focusing light microscopy [22.65]. In this work, the intensity distribution of the depleting light was formed by a 4Pi setting with destructive interference at the geometrical focus leading to an intensity zero there and two neighboring maxima at a distance of approximately \({\lambda}/4\). This resulted in superior \(xz\)-images, and the technique was initially successfully applied to membrane-labeled bacteria [22.65]. STED 4Pi microscopy was quickly extended to immunofluorescence imaging (Fig. 22.11a-f), and a spatial resolution of \(\approx{}{\mathrm{50}}\,{\mathrm{nm}}\) was demonstrated in the imaging of the microtubular meshwork of a mammalian cell in 2003 [22.75]. These results indicated that the basic physical obstacles had been overcome towards attaining a 3-D resolution of the order of a few tens of nanometers. Since the samples were mounted in an aqueous buffer [22.65, 22.75], the results indicated that the optical conditions for obtaining sub-diffraction resolution are met under the physical conditions encountered in live-cell imaging.

Fig. 22.11a-f

Sub-diffraction immunofluorescence imaging with STED 4Pi microscopy. (a) Overview image (\(xy\)) of the microtubular network of a human embryonic kidney (HEK ) cell. (b) Sketch of typical dimensions of a labeled microtubule fluorescently decorated via a secondary antibody. (c) Standard confocal and (d) STED 4Pi \(xz\)-image recorded at the same site of the cell; the straight line close to the cell stems from a monomolecular fluorescent layer attached to the adjacent cover slip. Pixel size \(95\times{}{\mathrm{9.8}}\,{\mathrm{nm}}\) in \(x\)- and \(z\)-direction, respectively; dwell time per pixel was \({\mathrm{2}}\,{\mathrm{ms}}\). The STED 4Pi microscope's PSF features two low side lobes caused by the secondary minima of the STED intensity distribution. These lobes are \(<{\mathrm{25}}\%\) and were removed in the STED 4Pi image using linear filtering. (e), (f) Corresponding profiles of the image data along the dashed lines in (c) and (d) quantify the improved axial resolution of the STED 4Pi microscopy mode over the confocal benchmark

Ultrasmall detection volumes created by STED also prove useful in a number of sensitive bioanalytical techniques. Fluorescence correlation spectroscopy ( ) [22.76] relies on small focal volumes to detect rare molecular species or interactions in concentrated solutions [22.77, 22.78]. While volume reduction can be obtained by nanofabricated structures [22.79], STED will prove instrumental in attaining ultrasmall spherical volumes at the nanoscale inside samples that do not allow for mechanical confinement. The latter fact is particularly important for avoiding an alteration in the measured fluctuations by the nanofrabricated surface walls.

The viability of STED FCS was first shown in a 2005 experiment [22.80]. In a particular implementation, STED FCS enabled a reduction in the focal volume by a factor of \(\mathrm{5}\) along the optical axis and concomitant reduction in the axial diffusion time. The initial experiments showed that for particular dye–wavelength combinations, the evaluation of the STED FCS data might be complicated by a seemingly uncorrelated background at the outer wings of the fluorescence spot where STED may not completely suppress the signal. Already then, published results implied the possibility of a further decrease in volume by another order of magnitude [22.28, 22.65], even while the combination of STED and FCS would prove most useful a few years later in the context of studying 2-D-confined molecular motions in cellular membranes. An inherent disadvantage of STED is the need for an additional pulsed light train that is tuned to the red edge of the emission spectrum of the dye. Nevertheless, to date, STED is the only known method for squeezing a fluorescence volume to the zeptoliter scale without making mechanical contact. Thus, the creation of ultrasmall volumes, tens of nanometers in diameter, by STED may be a pathway to improving the sensitivity of fluorescence-based bioanalytical techniques [22.81, 22.82].

An important step towards widespread application of STED microscopy was the demonstration of the suitability of laser diodes for both excitation and depletion [22.74]. However, several issues remained to be addressed. Because of the considerably smaller detection volumes, the signal per pixel is reduced and the number of pixels to be recorded increases. Therefore, it was clear early on that it would be important to incorporate STED into fast, ideally parallelized scanning systems.

It was also clear that the ultimate resolution limits in STED imaging would most likely be set by the stability of the marker used. The photostability of presently available markers was improved considerably by stretching the depleting pulse to \(> {\mathrm{300}}\,{\mathrm{ps}}\) [22.72], and longer pulses have since improved it further.

As explained above (22.30), the actual depth of the zero co-determines the attainable resolution, because for relatively high saturation factors, the saturable transition also becomes effective at the zero point or points. Typical depths have initially been in the range of \({\gamma}=1{-}2.5\%\) of the global maximum of the depleting intensity \(I(r)\), with considerable improvements realized since. The zero could be a single point, as in a single-beam scanning system, but in the case of a parallelized system, it may also be a line or an array of points or lines. The actual depth of the zeros certainly depends on the particular setup and the quality of optical components and proper alignment. Independently of implementation details, active optical elements such as wave-front phase modulators have since proven to be valuable tools for further deepening the zeros, which in turn enables one to better exploit the full potential of the attained saturation level for improved resolution.

Modern STED (or GSD, see below) microscopy (Figs. 22.12a-h22.17) has advanced well to the regime of a few tens-of-nanometer resolution, with application to cell-biological problems and beyond rapidly gaining momentum. The tunability of resolution, as seen in Fig. 22.12a-h, is a useful feature of STED/RESOLFT imaging, as the resolution can be matched to the objects imaged, a factor which becomes especially relevant when quickly sampling dynamic processes (see below).

Fig. 22.12a-h

Tunable resolution enhancement realized by STED microscopy. (a) Confocal and (b) STED image of fluorescent beads with average size of \(\approx{}{\mathrm{24}}\,{\mathrm{nm}}\) on a cover slip. (c–g) The area of the white rectangle shown in (a) and (b) recorded with different STED intensities. The resolution gain can be directly appreciated. (h) STED depletion \({\eta}\) versus STED light intensity measured on the same sample. The intensity settings for the measurements (c–g) are marked by arrows. Adapted with permission from [22.83], The Optical Society

Fig. 22.13a-d

Far-field fluorescence nanoscopy of the interior of integral cells. (a) Confocal microscopy overview of a mammalian (PtK2) cell outlining the mitochondrial network (gray) and the nucleus (blue). (b) Sketch of isoSTED nanoscope optically dissecting the interior of a mitochondrion with a nanospherical effective PSF. (c) isoSTED nanoscopy resolves and correctly identifies Tom20 (translocase of the outer mitochondrial membrane) protein complexes at the outer mitochondrial membrane. The complexes represent the sites where cellular proteins enter the mitochondria. (d) Nuclear pore complex architecture in an intact cell nucleus imaged by confocal microscopy (lower-left corner) and STED nanoscopy (remaining image field). The resolution of STED imaging is increased by about an order of magnitude (in the red channel), and eight molecular subunits of the nuclear pore complex can be resolved. (a,c) From [22.84], (d) Reprinted from [22.85], with permission from Elsevier

Fig. 22.14a-c

In situ access to 3-D morphological information: Examples from materials science: STED microscopy provides in-situ access to 3-D morphological information of nanostructured block copolymers. Transmission electron microscope ( ) image (a) confirming the lamellar morphology of a polystyrene-block-poly(2-vinylpyridine) polymer. Whereas the confocal reference (b) provides no details, scanning with an oblate STED 4Pi PSF ((b) left) faithfully reveals the observed nanostructure. (c) More isotropic morphologies, such as the bicontinuous structure shown, are revealed by means of a spherical isoSTED PSF. From [22.84]

Fig. 22.15a-d

Cross-sectional image (a) of a mesoporous morphology induced by selectively swelling the vinyl-pyridine phase subsequent to quaternization. The unzipped layers lining the pores (circled) are on average over half the size of the intact ones. (b) Perspective views of the corresponding binarized data stack. (c) Close-up of a selected area showing unzipping of the swollen domains which line the pores. (d) Visualization of a helicoid screw dislocation. From [22.84]

Fig. 22.16

Mixture of three \({\mathrm{200}}\,{\mathrm{nm}}\)-diameter fluorescent bead species in close packing with three different emission spectra (colors). Resolution in STED images: \(<{\mathrm{35}}\,{\mathrm{nm}}\) (green) and STED \(<{\mathrm{25}}\,{\mathrm{nm}}\) (red, blue). A confocal comparison is shown in the lower-right image portion

Fig. 22.17

Controlling the transfer of electrons to the conduction band (on state) in a coordinate-targeted manner by transferring electrons out of the valence band off state, as originally suggested in the classical ground state depletion (GSD) concept [22.10, 22.86, 22.87] and later also used in saturated structured-illumination microscopy ( ) [22.27, 22.88], allows one to separate sub-diffraction features in imaging based even on the short-lifetime photoluminescence ( ) from semiconductor heterostructures (here: GaP/GaInP segments in nanowires). Reprinted with permission from [22.89]. Copyright 2017 American Chemical Society

STED microscopy has been applied to study the fate of synaptic vesicle proteins during exocytosis. The study revealed that when the vesicles fuse with the presynaptic membrane, the protein synaptotagmin I forms integral nanosized clusters at the synapse [22.71]. Since the clusters are similar in size and molecular density to integral vesicles, the study indicated that entire protein clusters are taken up from the neuronal membrane when forming new vesicles. By the same token, this initial application of far-field fluorescence nanoscopy to a biological problem demonstrated the potential of this emerging field for solving long-standing problems in the life sciences. In another application, STED microscopy revealed the ring-like distribution of the Bruchpilot protein at synaptic active zones in the Drosophila neuromuscular junction [22.90]. Further studies included visualization of the spatial distribution of the SNARE (soluble N-ethylmaleimide-sensitive-factor attachement receptor) protein syntaxin [22.91], the nuclear protein SC35 [22.92], and the nicotinic acetylcholine receptor [22.93]. IsoSTED microscopy resolved the tube-like 3-D distribution of the TOM20 protein complex in the mitochondria in a mammalian cell [22.94]. The viability of STED microscopy with living cells was demonstrated in its early stages [22.68]. However, focusing on optical aspects of the concept, those experiments were carried out with slow piezo-scanning stages. Subsequent fast galvanometer scanning implementation of STED microscopy [22.95] visualized the rapid motion of dense synaptic vesicles at the synapse of living hippocampal neurons at video rate [22.96]. These results showed that nanoscale resolution and the visualization of rapid physiological processes can be reconciled, on the basis of existing physical principles and with available technology.

The nanosized detection area \({\Updelta}r\) or volume created by STED also extends the power of fluorescence correlation spectroscopy (FCS) and the detection of molecular diffusion [22.80, 22.97]. For example, STED microscopy has probed the diffusion and interaction of single lipid molecules on the nanoscale in the membrane of a living cell. The up to \(\approx 70\) times smaller detection areas created by STED (as compared to confocal microscopy) revealed marked differences between the diffusion of sphingo- and phospholipids [22.97]. While phospholipids exhibited a comparatively free diffusion, sphingolipids showed a transient (\(\approx {\mathrm{10}}\,{\mathrm{ms}}\)) cholesterol-mediated ‘‘trapping'' taking place in a \(< {\mathrm{20}}\,{\mathrm{nm}}\)-diameter area, which disappeared after cholesterol depletion [22.97, 22.98, 22.99] (Fig. 22.25a,b).

Stimulated emission occurs in all dyes investigated. However, just as in experiments using single molecules, the use of a number of dyes will be precluded by bleaching. Nonetheless, several suitable organic dyes have been found in each part of the spectrum, and new ones are continuously identified. In any case, the general demand for increased photostability is leading to the design of new labels with increased potential for STED microscopy, including fluorescent proteins. STED on yellow fluorescent proteins [22.100] was used early on in an application that quantified morphological changes in dendritic spines in living organotypic hippocampal brain slices upon external stimulation [22.101].

STED microscopy has important applications outside biology as well. For example, it is currently the only method that can be used to locally and noninvasively resolve the 3-D assembly of packed nanosized colloidal particles [22.102, 22.103] (compare also Fig. 22.16). In the realm of solid-state physics, STED microscopy has been used to image densely packed fluorescent color centers in crystals, specifically charged nitrogen vacancy (NV ) centers in diamonds [22.104]. NV centers in diamonds have attracted attention because of their potential application in quantum cryptography and computation, but also for nanoscale magnetic imaging. Since NV centers do not bleach or blink upon excitation, nanoparticles of diamond containing NV centers are also being developed as non-bleaching labels for bioimaging [22.105]. Figure 22.18a-f shows that these centers enable a virtually ideal implementation of the STED concept. The population of the bright state decreases with the intensity I, as one would expect from theory: \(\exp(-I/I_{\mathrm{S}})\). If \(I\gg I_{\mathrm{S}}\), the linear representation of the exponential fluorescence ‘‘depletion curve'' appears to be ‘‘rectangular'' [22.9], meaning that one can have a very narrow intensity range \(I<I_{\mathrm{S}}\) in which the NV centers are on and a broad intensity range \(I\gg I_{\mathrm{S}}\) where the center is off. As a result, STED microscopy was capable of imaging NV centers with a resolution of \({\Updelta}r=16{-}18\,{\mathrm{nm}}\) in raw data. Once separated by STED, the position of the NV centers could then be calculated with Ångström precision. Recording could be continually repeated without degradation of resolution or signal, demonstrating far-field nanoscale imaging without photobleaching. At the same time, these results underscore the potential of NV-containing diamond nanoparticles for biolabeling. Last but not least, increasing \(I\) yielded \({\Updelta}r={\mathrm{5.8}}\,{\mathrm{nm}}\), demonstrating an ‘‘all-physics-based'' resolving power exceeding the wavelength of light by two orders of magnitude [22.104]. These experiments corroborate the prediction of the original paper [22.9] that ‘‘rectangular depletion curves'' would allow the far-field resolution to be increased to the molecular scale.

Fig. 22.18a-f

STED microscopy reveals densely packed charged nitrogen vacancy (NV) color centers in a diamond crystal: (a) State diagram of NV centers in diamond showing the triplet ground (\({}^{3}\)A) and fluorescent state (\({}^{3}\)E) along with a dark singlet state (\({}^{1}\)E) and the transitions of excitation (Exc), emission (Em), and stimulated emission (STED). (b) The steep decline in fluorescence with increasing intensity shows that the STED beam is able to switch off the centers almost in a digital-like fashion. This nearly rectangular excited state depletion curve testifies to a close-to-ideal implementation of the STED effect. The inset confirms the exponential optical suppression of the excited state. For \(I_{\mathrm{STED}}> {\mathrm{20}}\,{\mathrm{MW{\,}cm^{-2}}}\), the NV center is in essence deprived of its ability to fluoresce, i. e., switched off. The on–off optical switching facilitates far-field optical separation of NV centers on the nanoscale. (c) The confocal image from the very same crystal region is blurred, while the STED image reveals individual NV centers. The spot produced by the individual centers represents the effective PSF of the STED recording. (d) An example of a \(y\)-profile of the PSF. (e) The location of each center can be calculated with ångström precision. (f) Data from a similar experiment, demonstrating a \(> 770\)-fold sharpening of the effective focal spot area through STED. Note that the increase in resolving power is a purely physical effect, i. e., just based on state transitions. From [22.104]

22.3.2 Variations of RESOLFT Microscopyand Producing Large SaturationFactors at Low Power

At this point, we reiterate that RESOLFT is not restricted to the process of stimulated emission, but can exploit any reversible (linear) transition driven by light; the attainable resolution is determined by the ratio of the driving intensity and the competing transition rate \(k_{\mathrm{BA}}\). If the applicable intensity is limited by the onset of photodamage to the marker, or even to the sample, marker constructs must be found where high saturation levels are attained at lower intensities (Figs. 22.1922.22a-c). This is certainly the case if the rate competing with the transition to be saturated is lower.

Fig. 22.19

The RESOLFT concept: Different molecular states and transitions can be applied to reversibly inhibit fluorescence for coordinate-targeted nanoscopy ranging from STED, over GSD, SPEM (Saturated Patterned Excitation Microscopy)/SSIM/GSD, to RESOLFT. From [22.106], reproduced with permission

Fig. 22.20a-d

GSD nanoscopy: (a) Dependence of the inhibition of fluorescence on the excitation laser intensity, which is based on transient shelving into a metastable dark state. (b) Pump-probe principle with pump light inducing dark state transitions and probe light exciting fluorescence of those molecules that are left in the bright state, resulting in a sub-diffraction observation spot. GSD and confocal (upper marked areas) images of immunolabeled SNAP-25 protein clusters on a fixed cell membrane (c) and an organic dye with a high triplet intersystem crossing rate, filling up a grooved nanostructure (d). From [22.106], reproduced with permission

Fig. 22.21a-e

RESOLFT nanoscopy using the reversibly photoswitchable protein rsGFP: rsGFP fulfills all requirements for coordinate-targeted nanoscopy: fast photoswitching ((a) fluorescence signal following repetitive on-off switching (red) with comparison to the reversibly switchable fluorescent protein ( ) Dronpa (blue)) with low switching fatigue ((bon fluorescence versus number of on–off switching cycle (red) and comparison to Dronpa (blue)), and a long lifetime of the off state ((c) spontaneous temporal recovery of fluorescence after off-switching with half of the fluorescence recovered after 23 min). RESOLFT and confocal images of an Escherichia coli bacterium expressing rsGFP-MreB (d) and a live mammalian cell expressing keratin-19-rsGFP (e). From [22.106], reproduced with permission

Fig. 22.22a-c

RESOLFT nanoscopy with more than \(\mathrm{100000}\) doughnuts: (a) By overlapping the diffraction pattern generated by two perpendicularly arranged gratings, an illumination pattern of the switch-off light is generated, which features a large number of intensity zeros ( : polarization beam-splitter, Obj.: objective lens) and which results in multiple simultaneous scanning points, whose dimension decreases with increasing intensity of the switch-off light (b). (c) Conventional (left panel) and RESOLFT (right panel) wide-field images of keratin 19-rsEGFP(N205S) in live mammalian cells. From [22.106], reproduced with permission

One such example is the ground state depletion (GSD) mentioned earlier. In this version of the RESOLFT concept, the ground state (now state A) is depleted by targeting an excited state (B) with a comparatively long lifetime [22.10, 22.11], such as the metastable triplet state \(\mathrm{T}_{1}\). In many fluorophores, \(\mathrm{T}_{1}\) can be reached through the \(\mathrm{S}_{1}\) with a quantum efficiency of \(1{-}10\%\) [22.107]. As a forbidden transition, the relaxation of the \(\mathrm{T}_{1}\) is \(E3{-}E5\) times slower than that of the \(\mathrm{S}_{1}\), thus yielding \(I_{\mathrm{S}}=0.1{-}100\,{\mathrm{kW/cm^{2}}}\). The signal to be measured (from the intensity zero) is the fluorescence of the molecules that remain in the singlet system; this measurement can be accomplished through a further synchronized excitation [22.10]. For many fluorophores, this approach is not straightforward, because \(\mathrm{T}_{1}\) is involved in the process of photobleaching, but there are potential alternatives, such as the metastable states of rare earth metal ions that are fed through chelates.

Studies have also proposed to deplete the ground state \(\mathrm{S}_{0}\) by populating the \(\mathrm{S}_{1}\) (now B) [22.27]. This is the technically simplest realization of saturated depletion, since it requires only excitation wavelength matching. However, as the fluorescence emission maps the spatially extended majority population in state B, the super-resolved images (represented by state A) are negative images hidden under a bright signal from B. Hence, photon noise from the large signal may swamp the fluorescence minima that occur when intensity zeros, where no fluorescence is excited, are co-localized with fluorophores. The subsequent computational extraction of the positive image is therefore highly dependent on an excellent signal-to-noise ratio. The saturation intensity is of the same order as in STED microscopy, because the saturation of fluorescence also competes against the spontaneous decay of \(\mathrm{S}_{1}\). This results in similar photostability issues as in the case of STED [22.108]. In fact, the photobleaching should be exacerbated, since the saturated transition is effected with higher-energy photons that are generally more prone to facilitating photochemical reactions. Pumping the dye to a higher state rather than into the ground state also favors photolability. Adding to the problem is the fact that a large number of dye molecules constantly undergo excitation-emission cycles in order to image a comparatively small spot. Finally, saturation of the \(\mathrm{S}_{1}\) will be possible only if the long-lived triplet state is not allowed to build up during repeated excitation. As most dyes feature a triplet relaxation rate of \(> 1/{\mathrm{1}}\,{\mathrm{\upmu{}s}}\) (that strongly depends on the environment), effective triplet relaxation requires a pulse repetition rate \(<{\mathrm{500}}\,{\mathrm{kHz}}\). Nevertheless, because of the simplicity of raw data acquisition, it may remain an attractive method for the imaging of very bright and photostable samples.

One possible solution to the quest for large saturation factors at low intensities is compounds with two (semi-)stable states [22.12, 22.65]. If the rate \(k_{\mathrm{BA}}\) (and the spontaneous rate \(k_{\mathrm{AB}}\)) almost vanish, large saturation factors are attained at very low intensities. The lowest useful intensity is then determined by the slowest acceptable imaging speed, which is ultimately determined by the switching rate. A favorable aspect is that in most bistable compounds, the speed of the actual switching mechanism, i. e., of the conformational change, is less than a few nanoseconds, which is much faster than the typical pixel dwell time in scanning. In the ideal case, the marker is indeed a bistable fluorescent compound that can be photoswitched at separate wavelengths, from a fluorescent state A to a dark state B and vice versa, where spontaneous rates will not influence this compromise.

Photoswitchable coupled molecular systems, such as an early example based on a photochromic diarylethene derivative and a fluorophore [22.107], are one attractive candidate for RESOLFT methods. Using the kinetic parameters reported, (22.27a) predicts that focusing of less than \({\mathrm{100}}\,{\mathrm{\upmu{}W}}\) of deep-blue switch-off light to an area of \({\mathrm{10^{-8}}}\,{\mathrm{cm^{2}}}\) for \({\mathrm{50}}\,{\mathrm{\upmu{}s}}\) should in principle yield better than \({\mathrm{5}}\,{\mathrm{nm}}\) spatial resolution. The targeted optimization of photochromic or other compounds towards fatigue-free switching and visible light operation is therefore expected to open up radical new avenues in microscopy and data storage [22.12], and some of this potential has clearly since been realized (see below).

For live-cell imaging, fluorescent proteins have many advantages over synthetic dyes. Many feature dark states with light-driven transitions [22.11, 22.12]. If the spontaneous lifetimes of these states are longer than \({\mathrm{10}}\,{\mathrm{ns}}\), such proteins may in principle permit much larger saturation factors. The most attractive solution, however, is fluorescent proteins that can be switched on and off at different wavelengths [22.12]. An early example was asFP595 [22.109]; insertion of the published data into (22.27a) predicted saturated depletion of the fluorescence state with intensities of less than a few \(\mathrm{W/cm^{2}}\) and, under favorable switching conditions, spatial resolutions of better than \({\mathrm{10}}\,{\mathrm{nm}}\) [22.12]. The intensities involved should also enable parallelization of saturation through an array of minima or dark lines. The initial realizations of very-low-intensity depletion microscopes were challenged by switching fatigue [22.107] and overlapping action spectra [22.109]. Nevertheless, the prospect of attaining nanoscale resolution with regular lenses and focused light has been seen as an incentive to overcome these challenges by strategic fluorophore modification [22.12], and these or similar types of fluorescent proteins [22.110] have been a good starting point for these efforts.

It is important to realize that while an effectively nonlinear interaction between light and fluorophore is the basis for this increased resolution, these methods do not require transitions involving more than one photon at a time, such as \(m\)-photon excitation, \(m\)-th harmonics generation, or coherent anti-Stokes-Raman scattering [22.66]. This means that the required intensities are not determined by the very small cross sections of these processes and the requirement for ultrahigh (peak) intensities. By contrast, the saturation of a linear optical transition depends on the basic kinetics of the population of the involved states. Consequently, pulse length requirements are less strict, and, most importantly, the required intensities can be significantly reduced by choosing appropriate spectroscopic systems.

The field is progressing rapidly, and some selected highlights include the demonstration of STED nanoscopy at millisecond imaging times for ultrafast dynamics in small fields of view [22.111], the demonstration of a RESOLFT strategy to neutralize the diffraction limit of light-sheet fluorescence microscopy [22.112], efficient STED nanoscopy with quantum dots as fluorescent labels [22.113], the highest levels of 3-D isotropic resolution (\(<{\mathrm{30}}\,{\mathrm{nm}}\) in \(x,y,z\) simultaneously) with a more stable design of 4Pi-based isoSTED [22.114], and the several-thousandfold massive parallelization of RESOLFT and even STED without resolution compromise for faster imaging of large fields [22.115, 22.116, 22.117] as well as expanded multicolor capabilities [22.118].

22.4 Coordinate-Stochastic State Transfer at the Single-Molecule Level: PALM, STORM, PAINT, and Related Approaches

The challenges posed by repeated cycling between molecular states for the coordinate-targeted STED/RESOLFT approaches are alleviated when transferring individual molecules [22.119, 22.120] between different states stochastically in space. For example, molecules that are initially off may be individually driven to their on-state at unknown spatial coordinates. The molecules' coordinates can be determined with sub-diffraction precision from their images on a camera. While the image of a single molecule is again blurred by diffraction, the molecular position can be determined by calculating the centroid of the blurred image spot. One restriction is that only single isolated molecules farther apart than the distance given by diffraction can be imaged at one time to avoid any bias in localization of molecular positions from overlapping (blurred) spots. In addition, molecules, once in their on-state, have to emit a sufficient number of photons N, since the localization precision scales with the inverse square root of \(N\) [22.128]. It is important, however, to realize that localization per se cannot provide super-resolution, i. e., finding a position of an object with arbitrary precision is not the same as resolution. Resolution is about separating similar objects at small distances. This is why, although it had routinely been applied for decades, specifically for spatiotemporal tracking of single isolated particles or molecules, localization on its own did not provide nanoscale images. Resolution requires a criterion to discern neighboring molecules such as realized by driving molecular transitions between different states [22.10, 22.129, 22.130, 22.9]. Therefore, an approach first suggested as (fluorescence) photoactivated localization microscopy ((F) ) [22.121, 22.131] or stochastic optical reconstruction microscopy ( ) [22.132] (Fig. 22.23a-fa,b) assembles a super-resolved image by determining spatial positions molecule by molecule using molecular transitions:
  1. 1)

    Only a few isolated molecules are stochastically transferred (or activated) into their on-state.

     
  2. 2)

    These molecules are imaged onto a camera, and their spatial coordinates are determined through localization and saved.

     
  3. 3)

    Molecules are transferred into an off-state.

     
  4. 4)

    Stochastic activation of another subset of isolated molecules allows the read-out of neighboring molecules.

     
  5. 5)

    Repetition of this cycle realizes the reconstruction of an image with sub-diffraction resolution from the spatial coordinates of all imaged molecules (Fig. 22.24, lower row).

     
Fig. 22.23a-f

Coordinate-stochastic nanoscopy: PALM, STORM, PAINT, and ground state depletion followed by individual molecule return ( ). (a) Comparative summed-molecule TIRF (left) and PALM (right) images of the same region within a cryoprepared thin section from a COS-7 cell expressing the lysosomal transmembrane protein CD63 tagged with the photoactivatable fluorescent protein Kaede. (b) 3-D STORM image of the mitochondrial network in a BS-C-1 cell (left), and dual-objective STORM image of actin (middle, labeled with Alexa Fluor 647 phalloidin) in a COS-7 cell. The \(z\) positions are color-coded (violet and red, positions closest to and farthest from substratum, respectively). (Right) Comparison of conventional (grayscale) and 3-D STORM images of actin in the axons of neurons. Actin is labeled with phalloidin conjugated to photoswitchable dyes. STORM imaging first revealed a periodic actin-spectrin-based membrane skeleton in axons. The PAINT concept: (c) Fluorescence image and reconstructed high-resolution image of vesicles attached to a glass surface. The reconstructed image displays the average coordinates of single fluorophores. Fluorescence and reconstructed images of two vesicles with a center-to-center distance of \({\approx}{\mathrm{200}}\,{\mathrm{nm}}\) reveal the enhanced resolution of PAINT. Lower row: Image of a supported bilayer on glass, probed by Nile red. (d) Spectrally multiplexed DNA -PAINT super-resolution imaging of microtubules and mitochondria inside fixed cells. GSDIM and related methods: (e) A fluorophore can be continuously cycled between its bright singlet and dark state system with a single excitation laser, effecting ON/OFF blinking of fluorescence in time and space. (f) GSDIM images of Rh6G-immunostained microtubules in PtK2 cells in aqueous buffer and of the microtubule cytoskeleton of living PtK2 cells labeled with the fluorescent protein Citrine-Map2. Upper left corners: diffraction-limited wide-field recordings. (a) From [22.121]. Reprinted with permission from AAAS, (b) Adapted with data from [22.122, 22.123, 22.124], (c) From [22.125]. Copyright (2006) National Academy of Sciences, USA, (d) From [22.126], (e,f) From [22.127]

Fig. 22.24

Targeted versus stochastic time-sequential readout of fluorophore markers of a nanostructured object within the diffraction zone whose lower bound is given by \({\lambda}/2n\). In the targeted readout mode, one of the two states is established at a sub-diffraction-sized spot at the position of a zero to read out an unknown number of fluorophore molecules. The image is assembled by deliberate translation of the zero. The zero can also be a groove. In the stochastic readout mode, a single switchable fluorophore from a random position within the diffraction zone is switched to a (stable) state A, while the other molecules remain in B. The coordinate is calculated from the centroid of the diffraction fluorescence spot measured by a pixelated detector. The coordinate pops up stochastically depending on where the interrogated marker molecule is located. After [22.129]

The original (F)PALM experiments employed photoactivatable fluorescent proteins [22.133], with the switch-on and switch-off accomplished using dedicated laser light and irreversible photobleaching, respectively [22.121, 22.131]. Similarly, photoactivatable organic dyes can be used [22.134, 22.135]. This, unfortunately, comes at the expense of the ability to record a molecule several times, i. e., to acquire structural changes in the specimen over time. The original STORM experiments applied reversible photoswitchable organic fluorophores such as cyanines [22.132]: under certain buffer conditions they can be transferred between a bright on-state and a dark off-state using red and green light, mediated by nearby activator fluorophores [22.136]. Photoswitching in cyanines and other organic dyes may also be accomplished by other molecular transitions such as via the transient population of metastable dark states including the dyes' triplet states or redox states populated therefrom [22.137, 22.138, 22.139, 22.140]. Similarly, RSFPs may be employed [22.120, 22.141, 22.142]. Harnessing reversible molecular transitions allows one to record a molecule's position several times, i. e., to acquire a sequence of super-resolution images [22.143, 22.144, 22.145, 22.146]. Alongside fluorescent proteins and organic dyes, (F)PALM/STORM-like recordings were realized with other emitters such as luminescent single-walled carbon nanotubes [22.147] or quantum dots ( s) [22.148].

The use of reversible molecular transitions led to the idea to continuously (and still stochastically) drive molecules between a bright and a dark state. Modalities termed PALM with independently running acquisition ( ) [22.141, 22.142] and ground state depletion followed by individual molecule return (GSDIM) [22.127] apply no activation or switch-on beam, and isolated fluorophores are allowed to blink stochastically and subsequently in time (not only in space) (Fig. 22.23a-fe,f) [22.127, 22.129]. A single continuous-wave laser beam is used to generate the \(N\) photons and to switch the fluorophores off by transferring them into dark states. Dark state return is promoted either by switching crosstalk of the laser or by a spontaneous decay. The camera is run freely and the laser intensity and frame rate adjusted such that the average duration of the \(N\)-photon burst coincides with the duration of a camera frame. These purely stochastic concepts are probably the simplest far-field nanoscopy systems, because they simply require uniform laser illumination, a freely running camera, and appropriate software. A straightforward advantage of such an acquisition mode is that it allows the use of conventional fluorophores such as many organic dyes or fluorescent proteins. Starting with STORM [22.132] and later GSDIM [22.127], experiments referred to as direct-STORM ( ) [22.138, 22.140] and blinking microscopy [22.139], or single-molecule active-control microscopy ( ) [22.149, 22.150], adapt buffer conditions and laser intensities to tune transitions to metastable dark states such as radical states of standard labels, producing super-resolution images of conventionally labeled samples or even of autofluorescent cellular structures [22.151]. The return from long-lived metastable states can often be accelerated with additional or infrared laser light, adding another parameter to optimize the acquisition of the single-molecule data [22.127].

Transient stochastic on-switching may also be effected by molecular collisions or chemical reactions, where a fluorophore is activated only once it is interacting with other specific molecules such as single-walled carbon nanotubes [22.147] or by probe binding as in the influential points accumulation for imaging in nanoscale topography ( ) concept (Fig. 22.23a-fc,d) [22.125]. Although a gamut of different notations have been introduced, they are all based on the same principle, namely, modulating the fluorescence emission of single molecules using molecular transitions. Differences appear only in the details of the experimental design, i. e., switching mechanism, how many lasers are used, the camera running mode, the choice of label, buffer conditions, etc.

As the PALM/STORM/PAINT approach is implemented in the wide-field, exciting opportunities exist to encode the axial positions of molecules in the diffraction patterns on the camera. A number of approaches are reviewed in [22.150]. These extend from simple astigmatic imaging [22.152] to more advanced schemes encoding positions over larger depth ranges, e. g., [22.153, 22.154]. Striving for optimal solutions, the 3-D localization problem can be put on a solid theoretical basis, and both the precision [22.155, 22.156] and other parameters such as emission color [22.157] or constraints to molecular orientation [22.158, 22.159, 22.160, 22.161, 22.162] can be encoded in ways that are compatible with super-resolved imaging.

22.5 Imaging Capabilities for a New Age of Nanobiology

After initial hurdles of limited access to the pioneering instruments that were available in only a small number of laboratories, adoption of fluorescence nanoscopy has picked up quickly, not least because of commercialization, a trend which is still accelerating. Nanoscopy has now been successfully used to probe the interior of both eukaryotic and prokaryotic cells, with a quickly increasing number of studies investigating new biological questions and surveying previously unattainable scales [22.163] (Fig. 22.25a,b).

Fig. 22.25a,b

STED microscopy discriminates the dynamics of single sphingo- and phospholipid molecules in a living cell membrane: Dye-labeled sphingomyelin (SM ) versus phosphoethanolamine (PE ). (a) Freely moving molecules may be transiently trapped in the membrane due to interaction with another molecule. The passage time of freely diffusing molecules through the small spot created by STED is substantially reduced compared to that through a diffraction-limited confocal microscope. Fluorescence bursts from labeled single PE and SM lipids detected with the \({\mathrm{250}}\,{\mathrm{nm}}\)-diameter confocal and a \({\mathrm{50}}\,{\mathrm{nm}}\)-diameter STED spot show that the smaller spot created by STED distinguishes free lipid diffusion (I) from interaction events (II). (b) The diffusion of PE and SM lipids on the plasma membrane quantified by fluorescence correlation spectroscopy. Normalized correlation data of PE (red) and SM (green) are compared for a standard confocal and STED recording depicting the heterogeneous diffusion of SM. While in the confocal case the difference between the SM and PE diffusion could be explained just by a generally slower diffusion of SM, the STED recording can only be explained by a transient trapping just of SM. Following depletion of cholesterol, the diffusion of SM (black, lowest curve) is similar to that of the rather freely diffusing PE. Thus, STED microscopy reveals cholesterol-dependent heterogeneous diffusion of SM on the nanoscale. Data adapted from [22.97]

Different areas of cell biology have adopted nanoscopy at different speeds. Whereas virology and microbiology, for example, were quick to use these approaches, in plant biology—a field with very strong roots in microscopy—only few nanoscopy studies have been reported thus far [22.164]. Initially, most nanoscopy studies utilized mammalian cells grown on cover glass. However, as the techniques and their availability developed, nanoscopy has been increasingly used to investigate nanoscale structures in cells grown in more complex environments such as 3-D cell cultures and biofilms, but also in tissues and even intact animals. Nanoscopy has been used to address questions in fields such as immunology [22.165], signaling [22.166], organelle structure [22.167], virology [22.168, 22.169, 22.170, 22.171, 22.172], bacteriology [22.173], and cancer biology [22.174]. Even the first reports on using nanoscopy on human patient tissue have appeared [22.175, 22.176], pointing to exciting future possibilities in medical diagnostics.

Nanoscopy [22.177] has also been applied to study the detailed architecture of molecular machines and complexes within cells. Notably, images of labeled nuclear pore complexes ( s) have been analyzed by single-particle averaging, a method well established in electron tomography, to map specific proteins within this structure [22.178, 22.179]. Based on the images of thousands of NPCs, it has been possible to determine the average positions of individual proteins of the Nup107-160 subcomplex with precision of well below 1 nm, shedding light on the molecular architecture of this pore [22.179]. This approach has even been transferred to reconstruct 3-D data sets [22.180], and similar strategies have been used to investigate symmetric structures such as primary cilia [22.181] or herpes simplex viruses [22.182]. Nanoscopy has now come of age, and its benefits typically manifest in combinations with other experimental approaches.

In the early days of nanoscopy, the fluorescently labeled cytoskeleton was used primarily as a convenient cellular structure to determine the resolving power of the instrument. It came as a surprise when it was shown, first in neuronal axons [22.122] and later also in dendrites [22.183, 22.184] and spine necks [22.185, 22.186], that short actin filaments capped by adducin are bridged by spectrin tetramers to form an \(\approx{}{\mathrm{190}}\,{\mathrm{nm}}\) periodic ring-like structure underneath the plasma membrane [22.122, 22.187, 22.188] (Fig. 22.26a-ea). This periodic lattice has now been visualized in both chemically fixed and living cells in virtually every neuron type and also in glial cells [22.189, 22.190, 22.191], suggesting that this is a more general feature of the neuronal cytoskeleton. The periodic scaffold forms a diffusion barrier [22.192] and regulates the positioning of other proteins, all exhibiting the same periodicity, including ankyrins, sodium and potassium channels, and adhesion molecules [22.122, 22.183, 22.187, 22.193]. This periodicity is even coordinated between axon and glial cells at the nodes of Ranvier [22.193] (Fig. 22.26a-eb). Remarkably, this periodic actin structure has so far eluded visualization by electron microscopy ( ). Hence this repetitive organization of the cytoskeleton in neurons, found by STORM and STED, is indeed a bona fide example of a subcellular structure discovered by fluorescence nanoscopy.

Fig. 22.26a-e

Fluorescence nanoscopy of neuronal cells: (a) Actin, spectrin, and other proteins form a coordinated quasi-1-D lattice structure in neuronal processes. (b) STED nanoscopy revealed that at paranodes (regions flanking the nodal gap), both axonal proteins and glial proteins form periodic quasi-1-D arrangements with a high degree of interdependence between the positions of the axonal and glial proteins. (c) Nanoscopy suggests that synaptic transmission is organized such that the active zone directs action potential-evoked vesicle fusion to occur preferentially at sites directly opposing postsynaptic receptor–scaffold ensembles. In this way, neurotransmitter release is aligned to the corresponding receptors in the postsynaptic cell along nanocolumns. Image of the presynaptic protein Rab3-interacting molecule (RIM ) (red) and the postsynaptic scaffolding molecule PSD-95 (blue). (d) Nanoscopy reveals that dendritic spine-neck plasticity regulates compartmentalization of synapses. Examples of fluorescently labeled dendritic spines and their structural plasticity during long-term potentiation. (e) Direct observations of fibrils of aggregation-prone mutant huntingtin exon 1 proteins and amyloid-\({\upbeta}\) peptides in cells. These fibrillar aggregates are formed from monomers and potentially higher-order oligomeric conformational species, which can be visualized by nanoscopy. From [22.163]

Since learning and memory is encoded in large part by the pre- and the post-synapses established by the transmitting and receiving neurons, an understanding of the functions of our brains requires detailed insights into the architecture of these structures. In an attempt to obtain a global view on the structure of the pre-synapse, synaptic boutons containing the entire synaptic machinery were purified from rat brains and analyzed by quantitative Western blotting, mass spectrometry, EM, and nanoscopy [22.194]. This led to the generation of an average synapse model, composed of the merged data sets. Although many details are missing from this model or need further verification, it gives a flavor of the enormous nanoscale complexity of a structure such as the synapse.

Thus far, most nanoscopy studies on synapses have concentrated on the presynaptic active zones (release sites) and the dendritic spines which contain the postsynaptic machinery [22.195, 22.196]. The protein-rich active zones are often smaller than the diffraction limit. A clearer picture of how molecular scaffolds and machineries are organized in sub-synaptic nanodomains is emerging, although the data are still far from providing a complete picture of all proteins [22.197, 22.198, 22.199, 22.200, 22.201, 22.202, 22.203, 22.204, 22.205, 22.206, 22.207, 22.71, 22.90]. Nanoscopy has contributed to the identification of long-range organization across the synaptic cleft, revealing molecular components that are co-aligned in the pre- and post-synapse. These so-called nanocolumn arrangements may facilitate effective synaptic communication [22.208] (Fig. 22.26a-ec).

Dendritic spines are specialized micron-sized membrane protrusions harboring the postsynaptic machinery. The spines contain a highly branched actin cytoskeleton network that influences their shape. Live-cell nanoscopy imaging revealed a high level of heterogeneity and pronounced dynamics of this network [22.209, 22.210, 22.211]. Remarkably, STED nanoscopy in combination with fluorescence recovery after photobleaching ( ) experiments and electrophysiology enabled the simultaneous measurement of spine neck geometry and the movements of molecules in and out of the spines, with high temporal resolution (Fig. 22.26a-ed) [22.101, 22.212, 22.213]. These data showed that biochemical compartmentalization of the spines separating them from the dendrites is critically dependent on the spine neck width [22.213]. This appears to be a major factor in the fine-tuning of synaptic strength, which is crucial for the computational power of neurons.

Neurodegenerative disorders such as Parkinson's, Alzheimer's, and Huntington's diseases are characterized by aberrant accumulations of proteins or peptides in the brain of patients, with subsequent neuronal cell death. A crucial and still unresolved question is which of the polymorphic aggregation states elicit toxic cellular responses and by which mechanisms. Nanoscopy is now at the stage to directly visualize protein aggregation, e. g., aggregate nucleation events and fibril formation [22.217, 22.218], and to carry out such studies in cells (Fig. 22.26a-ee) [22.219, 22.220, 22.221]. We expect that the ability to selectively co-visualize the action of cellular protein quality control and degradation systems, such as the ubiquitin-proteasome system, autophagy-mediated degradation and chaperone networks will add new information about which of the aggregation species are resilient to degradation and drive toxic cascades. As an example of the new capabilities of nanoscopy, it has been demonstrated that exogenous addition of a chaperonin subunit attenuates the formation of both huntingtin (Htt) inclusion bodies and other smaller fibrillar Htt species present in the cytosol of neuronal model cells [22.222, 22.223], potentially opening pharmaceutical avenues to interfere with these aggregation processes. Nanoscopy also provided information on the interactions of larger Htt aggregates with transcription factors, suggesting that these aggregates exert some of their cytotoxic effects by interfering with gene transcription [22.224].

In thicker tissue slices, and to an even greater extent in intact organisms, optical readout is complicated by sample-induced optical aberrations related to enhanced light absorption and scattering with increasing depth of imaging. Strategies from adaptive optics have been explored to alleviate these problems for both STED [22.225, 22.226] and coordinate-stochastic nanoscopy [22.227], though this is still a frontier. In practical terms, careful refractive index matching to the sample is an important first step to reducing aberrations in vivo [22.228]. The growing number of live-cell studies on single cells using various nanoscopy technologies clearly shows that nanoscopy is indeed a powerful and urgently needed tool [22.144, 22.189, 22.229, 22.230, 22.231]. However, the number of nanoscopy studies focusing on living tissues and intact organisms is still limited to a few proof-of-concept studies, all of which use coordinate-targeted schemes [22.101, 22.210, 22.213, 22.232]. Examples of nanoscopy in whole animals (the mouse and fruit fly larvae) are displayed in Figs. 22.27a-c22.29 [22.214, 22.215, 22.216].

Fig. 22.27a-c

STED nanoscopy of a mouse with enhanced yellow fluorescent protein-labeled neurons. Shown are dendritic and axonal details in the molecular layer of the somatosensory cortex of a living, anesthetized mouse. (a) Optical access to the brain cortex was enabled by a cover glass-sealed cranial window. (b) Image of a neuron. (c) STED time-lapse recording of spine morphology dynamics. From [22.214]. Reprinted with permission from AAAS

Fig. 22.28a,b

STED imaging of synaptic protein distribution. Example: PSD95, the abundant scaffold protein at the postsynaptic membrane, which organizes numerous other synaptic proteins. (a) The in vivo labeling of endogenous PSD95-HaloTag, a self-labeling enzymatic protein tag, with organic fluorophores. (b) Depending on the orientation of the individual spine head imaged with respect to the focal plane, the intricate spatial organization of PSD95 at the synapse is revealed in the STED mode. From [22.215]

Fig. 22.29

RESOLFT imaging of the microtubule cytoskeleton of intact, living Drosophila melanogaster larvae. A second instar larva ubiquitously expressing a fusion protein composed of the RSFP rsEGFP2. From [22.216] published under CC-BY license

22.6 Nanoscopy at the MINimum: Molecular Resolution with MINFLUX

Improvements to STED microscopy have expanded its capabilities substantially to include an increasingly diverse range of cell-biological applications [22.163]. Specifically, recent adaptive scanning strategies [22.233, 22.234, 22.235] have proven key to reducing the overall light dose applied to the sample. These conceptual additions to STED/RESOLFT imaging reduce photobleaching [22.108] and are advantageous for live-cell imaging. Thus, they have allowed the resolution of STED microscopy to be pushed even closer to the \(< {\mathrm{20}}\,{\mathrm{nm}}\) regime for organic fluorophores and for routine users, under realistic cell-imaging conditions. The first example of these approaches is MINFIELD [22.233], which provides major signal increases and prolonged acquisitions by restricting imaging to regions below the diffraction limit (Fig. 22.30a-c). MINFIELD STED microscopy avoids the exposure of the fluorophore to excess intensities of the doughnut and, more generally, to the maxima of the light intensity distribution used for on/off-switching. Rapid and repetitive MINFIELD recording is likely to be the approach of choice for investigating small spatial domains, such as within the synapse. Moreover, MINFIELD STED microscopy will allow fast dynamic nanoscale processes to be captured on millisecond timescales and beyond [22.111, 22.233].

Fig. 22.30a-c

The MINFIELD concept: Lower local de-excitation intensities in STED nanoscopy for image sizes below the diffraction limit. (a) In STED imaging with pulsed lasers, the ability of a fluorophore to emit fluorescence decreases nearly exponentially with the intensity of the beam de-exciting the fluorophore by stimulated emission. \(I_{\mathrm{S}}\) can be defined as the intensity at which the fluorescence signal is reduced by 50%. Fluorophores delivering higher signal are defined as on, whereas those with smaller signal are defined as off. (b) The STED beam is shaped to exhibit a central intensity zero in the focal region (i. e., a doughnut), so that (c) molecules can show fluorescence only if they are located in a small area in the doughnut center. This area decreases with increasing total doughnut intensity. Due to its diffraction-limited nature, the intensity distribution of the STED focal beam extends over more than half of the STED beam wavelength and exhibits strong intensity maxima, significantly contributing to photobleaching. By reducing the size of the image field to an area below the diffraction limit, where the STED beam intensity is more moderate, one can reduce the irradiation intensities in the area of interest, inducing lower photobleaching and allowing the acquisition of more fluorescence signal at higher resolution. After [22.233]

DyMIN is a related novel recording strategy [22.235] which minimizes exposure to unduly high intensities except at scanning steps where these intensities are strictly required for resolving features (Figs. 22.31a-d and 22.32a,b). Like MINFIELD, the DyMIN approach achieves dose reductions by up to orders of magnitude, particularly for relatively sparse fluorophore distributions. Initially demonstrated for STED immunofluorescence imaging, both MINFIELD and DyMIN will be explored for other classes of fluorophores, including the inherently lower-light-level RESOLFT nanoscopy variants with genetically encoded fluorescent proteins. The recently described organic switchable photochromic compounds [22.236] will also be further developed as attractive alternatives in this regard. The synergistic combination of two separate fluorophore state transitions in a recent concept, multiple off-state transitions ( ) for nanoscopy [22.234], has also enabled many more image frames to be captured, at much improved contrasts and with lower STED light dose at a given resolution than for standard STED. Approaches for directly counting molecules with STED have also been developed [22.237], and can be used to quantify the composition of suitably labeled molecular clusters.

Fig. 22.31a-d

DyMIN STED imaging: Concept illustrated for two fluorophores spaced less than the diffraction limit. (a) Signal is probed at each position, starting with a diffraction-limited probing step (\(P_{\mathrm{STED}}\) = 0), followed by probing at higher resolution (\(P_{\mathrm{STED}}\) > 0). At any step, if no signal indicates the presence of a fluorophore, the scan advances to the next position without applying more STED light to probe at higher resolution. (b) For signal above a threshold, the resolution is increased in steps (c), with decisions taken based on the presence of signal. This is continued up to a final step of \(P_{\mathrm{max}}\) (full resolution where required). For the highest-resolution steps, directly at the fluorophore, the probed region itself is located at the minimum of the STED intensity profile (d). After [22.235]

Fig. 22.32a,b

Dual-color isotropic nanoscopy of nuclear pore components and lamina with DyMIN STED: (a) Confocal and 3-D DyMIN STED recordings of nuclear pore complexes (green) and lamina (red). (b) DyMIN STED imaging of DNA origami structures with fluorophore assemblies. The DNA origami-based nanorulers with nominally \({\mathrm{30}}\,{\mathrm{nm}}\) separation (\({\mathrm{10}}\,{\mathrm{nm}}\) gap) consisted of two groups of on average \(\approx{}15\) ATTO 647N fluorophores each. Accounting for the known \(\approx{}{\mathrm{20}}\,{\mathrm{nm}}\) extent of the fluorophore groups, the widths of the Gaussians imply an effective PSF of \(\approx{}{\mathrm{17}}\,{\mathrm{nm}}\) (FWHM). After [22.235]

It should be noted that a bright pattern of emitted light is required to ascertain the position of emission in PALM/STORM/PAINT…, just as a bright pattern of incident light ist needed in STED/RESOLFT to determine the position of emission (Figs. 22.33 and 22.34). Not surprisingly, bright patterns of light are always needed when it comes to positions, because if one has just a single photon, this on its own tells nothing on position. The photon can go anywhere within the realm of diffraction, there is no way to control where it goes within the diffraction zone. In other words, when dealing with positions, by definition one needs many photons, because this is inherent in diffraction. Many photons are required for defining positions of on- and off-state molecules in STED/RESOLFT microscopy, just as many photons are required to determine the position of on-state molecules in the stochastic PALM method. One is not confined to using a single doughnut (a single diffraction zone) in STED/RESOLFT. A wide-field arrangement can be used, meaning that we can also record a large field of view. To this end, we parallelize the scanning using an array of intensity minima, such as an array of doughnuts [22.115, 22.117]. Again, the fundamental difference from the spatially stochastic methods is that the positions where the molecules can assume the on or the off state are tightly controlled by the pattern of light with which we illuminate the sample. This is true regardless of whether there is one molecule at the intensity minimum of the pattern, or three molecules; however many, it does not matter.

Fig. 22.33

(a) In the coordinate-targeted mode, the coordinates of the on state are established by illuminating the sample with a pattern of light featuring an intensity zero; the location of the zero and the pattern intensity define the coordinates with sub-diffraction precision. (b) In the coordinate-stochastic mode, the coordinates of the randomly emerging on-state molecules are established by analyzing the light patterns emitted by the molecules (localization). Precision of the spatial coordinate increases in both cases with the number of photons in the patterns, i. e., by the intensity of the pattern. In both families of methods, neighboring molecules are discerned by transiently creating different molecular states in the sample

Fig. 22.34

To parallelize STED/RESOLFT scanning, a wide-field arrangement with an array of intensity minima may be used. The numbers of molecules at these readout target coordinates do not matter, while PALM etc. requires that there may be only a single on-state molecule within a diffraction zone, i. e., within the distance dictated by the diffraction barrier. The position of each on-state molecule is, however, completely random in space. \(I_{\mathrm{S}}\) can be regarded as the number of photons that one needs to ensure that there is at least one photon interacting with the molecule, pushing it from one state to the other in order to create the required difference in molecular states. \(I/I_{\mathrm{S}}\) is, so to speak, the number of photons which really elicit the (on/off)-state transition at the molecule, while most of the others just pass by. Similarly, in the PALM concept, the number of photons \(n\) in \(1/{\sqrt{n}}\) is the number of those photons that are really detected at the coordinate-giving pixelated detector (camera), i. e., that really contribute to revealing the position of the emitting molecule. From [22.238]

Although the PALM principle can also be implemented on a single diffraction zone (i. e., using a single focused beam of light), it is usually implemented in a parallelized way, that is, on a larger field of view containing many diffraction zones. PALM parallelization requires that there may be only a single on-state molecule within a diffraction zone, i. e., within the distance dictated by the diffraction barrier. However, the position of this molecule is completely random. Therefore, we must make sure that the on-state molecules are farther apart from each other than the diffraction barrier, so that they are still identifiable as separate molecules. While in STED/RESOLFT the position of a certain state is given by the pattern of light falling on the sample, the position in PALM is established from the pattern of (fluorescence) light emanating from the sample.

What does \(I/I_{\mathrm{S}}\) in STED/RESOLFT represent? \(I_{\mathrm{S}}\) can be seen as the number of photons that one needs to ensure that there is at least one photon interacting with the molecule, pushing it from one state to the other in order to create the required difference in molecular states. \(I/I_{\mathrm{S}}\) is, so to speak, the number of photons which really can do something at the molecule, while most of the others just pass by. Similarly, in the PALM concept, the number of photons \(n\) in \(1/{\sqrt{n}}\) is the number of those photons that are detected, that is, that really contribute to revealing the position of the emitting molecule. In other words, in both concepts, to attain a high coordinate precision, one needs many photons that really do something. This analogy very clearly shows the importance of the number of photons to achieve coordinate precision in both concepts. However, in both cases the separation of features is, of course, accomplished by an on/off transition.

Of all the nanoscopy or super-resolution advances of the last decade, the recently described (nanoscopy with minimal photon fluxes) concept [22.239] stands out, because it contains a radically new idea. Whereas in PALM/STORM the localization of a molecule is based on maximizing the number of detected fluorescence photons on a camera, which is inevitably limited by bleaching, in MINFLUX (Figs. 22.35a-d22.39a,b) the molecule is localized by making it coincide with the intensity zero of a doughnut-shaped excitation beam. The excitation beam is scanned across the molecule and the fluorescence is typically recorded as in a confocal microscope. The position of the molecule is ultimately identical to the position of the doughnut at which fluorescence emission is minimal (Fig. 22.35a-d). By fundamentally reducing the number of detected photons required for nanometer-precise localization, MINFLUX has opened the door to low-light-level optical analysis of tiny objects at true molecular scale resolution (\(1{-}5\,{\mathrm{nm}}\)). With MINFLUX, lens-based fluorescence microscopy has thus reached the ultimate resolution limit: the size of the fluorescent molecule itself. Moreover, the resolution is attained at relatively high speed, at least 10 times that in PALM/"​"​STORM.

While the experimental developments of the MINFLUX concept are still in the nascent stage, it is worth commenting on the fundamental advantage over localization based on the emitted fluorescence alone. As discussed in [22.239, 22.240], in PALM/STORM, as in camera-based tracking applications, a molecule's position is inferred from the maximum of its fluorescence diffraction pattern (back-projected into sample space). The precision of such camera-based localization ideally reaches \({\sigma_{\mathrm{cam}}}\geq{\sigma}_{\mathrm{PSF}}/{\sqrt{N}}\), with \({\sigma}_{\mathrm{PSF}}\) being the standard deviation of the pattern and \(N\) the number of fluorescence photons making up the pattern [22.128]. Note that \({\sigma}_{\mathrm{cam}}\) is thus clearly bounded by the finite fluorescence emission rate, which for fluorophores in current use rarely allows more than a few hundred photon detections per millisecond (\(<{\mathrm{1}}\,{\mathrm{MHz}}\)). Moreover, emission is frequently interrupted and eventually ceases due to blinking and bleaching. This also keeps the photon emission rate as the limiting factor for the obtainable spatiotemporal resolution. As a result, state-of-the-art single-molecule tracking performance long remained in the range of tens of nanometers per several tens of ms. Drawing on the basic ideas of the coordinate determination employed in STED/RESOLFT microscopy, the MINFLUX concept addresses these fundamental limitations [22.239]. By localizing individual emitters with an excitation beam featuring an intensity minimum that is spatially precisely controlled, MINFLUX takes advantage of coordinate targeting for single-molecule localization. The basic steps are illustrated for one spatial dimension in Fig. 22.35a-d. In a typical two-dimensional MINFLUX implementation, the position of a molecule is obtained by placing the minimum of a doughnut-shaped excitation beam at a known set of spatial coordinates in the molecule's proximity. These coordinates are within a range \(L\) in which the molecule is anticipated (Fig. 22.36a-d). Probing the number of detected photons for each doughnut minimum coordinate yields the molecular position. It is the position at which the doughnut would produce the minimal emission if the excitation intensity minimum were targeted to it directly. As the intensity minimum is ideally zero, it is the point at which emission is ideally absent. The precision of the position estimate increases with the square root of the total number of detected photons and, more importantly, by decreasing the range \(L\), the spatial scale inserted from the outside into the experiment. For small ranges \(L\), for which the intensity minimum is approximated by a quadratic function, the localization precision does not depend on any wavelength, and for the case of no background and perfect doughnut control, the precision \(\sigma_{\mathrm{MINFLUX}}\) simply scales with \(L/{\sqrt{N}}\) at the center of the investigated range. In other words, the better the coordinates of the excitation minimum match the position of the molecule, the fewer fluorescence detections are needed to reach a given precision. In the conceptual limit where the excitation minimum coincides with the position of the emitter, i. e. \(L=0\), the emitter position is rendered by vanishing fluorescence detection. This is contrary to conventional centroid-based localization, where precision improvements are tightly bound to having increasingly larger numbers of detected photons (compare Fig. 22.37).

Fig. 22.35a-d

Principles of MINFLUX, a concept for localizing photon emitters in space, illustrated in a single dimension using a standing light wave. (a) The unknown position x\({}_{\mathrm{m}}\) of a fluorescent molecule is determined by translating the standing wave such that one of its intensity zeros travels from \(x=-L/2\) to \(L/2\), with \(x_{\mathrm{m}}\) somewhere in between. (b) Solving \(f(x_{\mathrm{m}})=0\) yields the molecular position \(x_{\mathrm{m}}\). Equivalently, the emitter can also be located by exposing the molecules to only two intensity values belonging to functions \(I_{0}(x)\) and \(I_{1}(x)\) that are fixed in space, having zeros at \(x=-L/2\) and \(L/2\), respectively. Establishing the emitter position can be performed in parallel with another zero by targeting molecules farther away than \({\lambda}/2\) from the first one. (c) Localization considering the statistics of fluorescence photon detection: Success probability \(p_{0}(x)\) for various beam separations \(L\) for \({\lambda}={\mathrm{640}}\,{\mathrm{nm}}\). The fluorescence photon detection distribution is shown along the right vertical axis of normalized detections. The distribution of detections is mapped into the position axis \(x\) through the corresponding \(p_{0}(x,L)\) function (gray arrows), delivering the localization distribution. The position estimator distribution contracts as the distance \(L\) is reduced. (d) Cramér–Rao bound ( ) for each \(L\). Precision is maximal halfway between the two points where the zeros are placed. For \(L={\mathrm{50}}\,{\mathrm{nm}}\), detecting just \(\mathrm{100}\) photons yields a precision of 1.3 nm. After [22.239]

Fig. 22.36a-d

Implementation of MINFLUX in 2-D fluorescence imaging and tracking. (a) Diagrams of the positions of the doughnut in the focal plane and resulting fluorescence photon counts. Basic application modalities of MINFLUX. (b) Nanoscopy: A nanoscale object features molecules whose fluorescence can be switched on and off, such that only one of the molecules is on within the detection range. They are distinguished by abrupt changes in the ratios between the different \(n_{0},n_{1},n_{2}\), and \(n_{3}\) or by intermissions in emission. (c) Nanometer-scale (short-range) tracking: The same procedure can be applied to a single emitter that moves within the localization region of size L. As the emitter moves, different fluorescence ratios are observed that allow the localization. (d) Micrometer-scale (long-range) tracking: If the emitter leaves the initial L-sized field of view, the triangular set of positions of the doughnut zeros is (iteratively) displaced to the last estimated position of the molecule. By keeping it around \(\boldsymbol{r}_{0}\) by means of a feedback loop, photon emission is expected to be minimal for \(n_{0}\) and balanced between \(n_{1}\), \(n_{2}\), and \(n_{3}\), as shown. After [22.239]

Fig. 22.37

With MINFLUX nanoscopy one can, for the first time, separate molecules optically which are only a few nanometer apart. PALM/STORM delivers a diffuse image of the molecules, and the position of the individual molecules can be easily discerned with the practically realized MINFLUX

The tracking of fluorophores with substantially sub-millisecond position sampling (Fig. 22.38) is only the beginning in a quest for the highest spatiotemporal capabilities (compare data in Fig. 22.39a,b) [22.240]. The inherent detection with a confocal pinhole should also provide a critical advantage when considering imaging in denser and three-dimensional specimens, such as brain slices and in vivo imaging scenarios. With further development of other aspects, including field-of-view enlargement, MINFLUX is bound to transform the limits of what can be observed in cells and molecular assemblies with light. This impact will most likely be realized in cell and neurobiology, and possibly structural biology as well. Moreover, it should be a great tool for studying molecular interactions and intra-macromolecular dynamics in a range never before accessible.

Fig. 22.38

Many much faster movements can be followed than is possible with STED or PALM/STORM microscopy. Movement pattern of 30S ribosomes (colored) in an E. coli bacterium (grayscale). Movement pattern of a single 30S ribosome (green) shown enlarged. After [22.239]

Fig. 22.39a,b

MINFLUX tracking of rapid movements of a custom-designed DNA origami. (a) Diagram of the DNA origami construct with a single ATTO 647N fluorophore attached at the center of the bridge (\({\mathrm{10}}\,{\mathrm{nm}}\) from the origami base). By design, the emitter can move on a half-circle above the origami and is thus ideally restricted to a 1-D movement. (b) Histogram of \(\mathrm{6118}\) localizations of the sample with \({\updelta}t={\mathrm{400}}\,{\mathrm{{\upmu}s}}\) time resolution and a \(1.5{\times}1.5\)-\(\mathrm{nm}\) binning. The predominant motion is along a single direction. (c) A \({\mathrm{300}}\,{\mathrm{ms}}\) excerpt of the photon count trace (time resolution \({\updelta}t={\mathrm{400}}\,{\mathrm{{\upmu}s}}\) per localization). From [22.240]

Notes

Acknowledgements

The authors thank all members of the Department of NanoBiophotonics, Max Planck Institute for Biophysical Chemistry, over the years for their contributions to this work and for valuable discussions. Parts of the chapter draw on previous texts from [22.106, 22.163]and the Nobel Lecture delivered by S.W.H. in Stockholm on December 8, 2014. A first version of this chapter, on which parts of the present chapter are based, was published in 2005 and reprinted in 2007.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Steffen J. Sahl
    • 1
    Email author
  • Andreas Schönle
    • 2
  • Stefan W. Hell
    • 3
  1. 1.Dept. of NanoBiophotonicsMax Planck Institute for Biophysical ChemistryGöttingenGermany
  2. 2.Abberior Instruments GmbHGöttingenGermany
  3. 3.Dept. of NanoBiophotonics/Dept. of Optical NanoscopyMax Planck Institute for Biophysical Chemistry & Max Planck Institute for Medical ResearchGöttingenGermany

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