For 4Pi microscopy, the sample is sandwiched between two coverslips. This configuration permits focusing the coaxial but antiparallel laser beams into a common point within the sample. Because the working distance of high NA objectives is very small, the layer thickness between the two coverslips should not exceed 30 μm. Furthermore, great care has to be taken to avoid a deformation of the 4Pi point spread function (PSF), which can be induced by any refractive index mismatch. It is therefore essential to use the highest-quality coverslips of well-defined thickness (see Note 2 ) and to adjust the correction collar of the objectives properly (see Note 3 ). Additionally, the sample has to contain a reference which can be used to align the focus and phase of the counter-propagating waves. This reference can consist of either fluorescently labeled endogenous point- or rodlike structures of sub-diffraction dimensions or of artificial immobilized fluorescent beads (see Note 4 ). To prepare such samples use the following protocol:

For oil immersion lenses in combination with 97 %, TDE as embedding medium (see Note 5 ), the best results have been reported by a stepwise increase of TDE concentration during several washing steps of the fixed specimen [26].

With glycerol immersion lenses, all components between the two opposing objective lenses have to be adjusted to a refractive index close to 1.46. This can be achieved by using quartz coverslips and immersing cells in 87 % PBS-buffered glycerol (see Note 6 ). Of course, quartz coverslips are much more costly than standard BK7 glass coverslips.

With water immersion lenses, samples can be mounted in aqueous buffer (n = 1.33–1.34), but special care has to be taken to avoid even the slight tilting of the coverslips.

To obtain full image resolution, 4Pi data have to be post-processed to remove ghost images. Various methods have been described for this purpose ranging from three- or five-point linear filters [32] to nonlinear mathematical image restoration algorithms [8]. Like in any imaging system, the image can be described by a convolution of the unknown object function with the PSF of the microscope. In a first approximation, the 4Pi PSF can be separated into independent radial and axial contributions allowing for fast deconvolution algorithms. Here, the axial profile is approximated by a convolution of a peak intensity function describing a single maximum and a lobe function representing relative position and height of the three maxima. Side lobes can be then removed by convolution of the image function with the inverse of the lobe function. This so-called three- or five-point linear deconvolution is fast and robust and does not require highly accurate PSF measurements. But two requirements have to be fulfilled to make this approach feasible. First, the relative height of the side lobes must be constant with respect to a lateral offset from the optical axis. Second, the PSF has to be spatially invariant in the image. When these requirements are met, linear point deconvolution enables easy side lobe removal without highly accurate PSF measurements, but at the expense of a decreased effective signal-to-noise ratio and no further enhancement of resolution.

In contrast, resolution can be further enhanced by direct linear deconvolution, i.e., inverse filtering of the complete image with the 4Pi PSF. This works well for noise-free images and a spatially invariant and perfectly known PSF. In reality, however, an additional filter (like, e.g., Wiener filter) has to be implemented to account for noise. Though an additional resolution enhancement up to 50 % can be achieved using linear deconvolution, a limited signal-to-noise ratio and improper description of the 4Pi PSF make this approach unfeasible in many cases.

Nonlinear image restoration methods, e.g., based on Richardson-Lucy deconvolution, use a maximum-likelihood algorithm to iteratively minimize the deviation between the measured image and estimated object. These algorithms offer the advantage of noise reduction and resolution enhancement but are slow and susceptible to poor PSF measurements.

One prerequisite for all deconvolution methods described here is a spatial invariant PSF in the image. In practice, this is not always the case, i.e., the PSF depends on the position in the sample most often manifested in a phase shift of the interference pattern along the optical axis. Methods for deconvolution with a variable PSF have been described to account for this problem [33].

The position of single point-like objects, although imaged as diffraction limited spots, can be determined with a precision that is one order of magnitude higher than the optical resolution by fitting the measured photon distribution to an ideal Gaussian [34]. In 4Pi microscopy, the main peak of the PSF has an FWHM of 100 nm in the direction of the optical axis at optimal conditions and thus is 6–7-fold smaller than that of a confocal PSF. The localization precision of diffraction-limited signals scales with ~FWHM/√n, with n being the number of photons detected, meaning that improved optical resolution, at least in principle, provides higher accuracy. Additionally, two opposing lenses are capable of detecting more photons than a single lens which leads to a further increase of localization accuracy. As the 4Pi PSF is intrinsically elliptical, a true 2D fitting routine is essential for proper centroid localization (see Note 7 ). Taking these parameters into account, a localization accuracy of only a few nm has been demonstrated for 4Pi microscopy [35, 36].

Labeling of mammalian cell nuclei with fluorescent NPC-specific antibodies typically results in a punctate staining of the nuclear periphery. The dots of the punctate pattern usually have a density of ~5/μm² and a corresponding nearest neighbor distance of ≤0.4 μm. By confocal scanning microscopy it has been shown that the dots predominantly represent single NPCs, while NPCs which occur as part of a cluster cannot be resolved [37, 38]. The central axis of the NPC is perpendicular to the plane of the nuclear envelope, and therefore the central axis of NPCs in the upper and lower nuclear membrane co-aligns with the optical axis of the microscope. All these parameters—size, density, and orientation—make the NPC an ideal object for 4Pi studies. We started 4Pi analysis of the NPC by using a primary antibody against Nup358, the major component of the cytoplasmic filaments of the NPC.

A comparison of confocal and 4Pi axial sections is shown in Fig. 2. In confocal images NPCs appear as diffraction limited spots which are blurred in xz-direction (Fig. 2a, d). In contrast, 4Pi images of NPCs consist of a narrow main maximum (FWHM ~ 110 nm) accompanied by two ghost images of ~30 % relative intensity (Fig. 2c, e). These ghost images could be removed by using simple 3-point linear deconvolution (see Note 8 ) resulting in a true resolution of 110 nm along the optical axis (Fig. 2f). Since the density of NPCs is relatively large in HeLa cells, the increase in resolution simplifies discrimination of single NPCs in 4Pi images. However, a close inspection of the representative optical section through a complete HeLa cell nucleus reveals that even using an almost isotropic refractive index throughout the optical path, the 4Pi PSF still depends on optical path length in the specimen (Fig. 2e). While in the lower nuclear membrane, the 4Pi raw images of the NPC consist of strong main peaks and weak side lobes, one of the side lobes is almost as strong as the main peak at the upper nuclear membrane, making the PSF asymmetric. Therefore, the applied deconvolution scheme is quite successful on the lower, but not on the upper nuclear envelope. An improvement in such cases could be achieved by active external phase compensation or more sophisticated deconvolution algorithms.