1. Introduction

Light-sheet microscopy (LSM) illuminates only that part of the object that is in the focal plane and therefore offers three major advantages: first, high contrast images are captured because useful information from the focal plane is not superimposed by defocused background light [1–3]; second, fluorophores are excited only in the focal plane, leading to strongly reduced photobleaching [4,5] and third, widefield detection enables high speed imaging by exploiting the high framerates of modern cameras [6]. Whereas these characteristics allow excellent imaging of nearly transparent objects like zebrafish [7], other objects like Drosophila melanogaster [8] or Arabidopsis root tips [9] make high-quality imaging deep inside the object more difficult, since scattering and absorption reduce the penetration depth and illumination homogeneity.

The effects of scattering and absorption could be reduced by the following 6 techniques: i) Two side illumination [10] does not correct artefacts, but doubles the maximum object size that can be imaged. ii) It is known that interference of scattered and unscattered laser light leads to inhomogeneous illumination often resulting in dark stripes [11]. By pivoting light-sheets these interference patterns are averaged out and the image quality is improved [10]. iii) Scanned light-sheets [12] are superior to static light-sheets, because interference artefacts can be strongly reduced. iv) Line-confocal detection [13,14] and structured illumination [15,16] reduce the influence of multiply scattered photons and thereby increase the image contrast. v) Adaptive optics have been applied to compensate for object induced aberrations [17–21]. vi) Bessel beams offer an increased penetration depth into scattering samples [22] and enhance the homogeneity of the illumination [23]. In addition, the resolution in detection direction can be improved since illuminating Bessel beams feature a higher numerical aperture (NA) at the same depth of field compared to Gaussian beams [24].

It is known that the main maximum of a Bessel beam is surrounded by a ring system, which carries a significant part of the total beam power and which illuminates out of focus regions, thus reducing image contrast. This detrimental effect is increased for higher NAs, such that an improved axial resolution and a more homogeneous illumination can only be achieved at the expense of a lower contrast. Different contrast enhancing techniques for Bessel beams have been proposed. Two-photon excitation [25], coherent superposition of multiple Bessel beams [26], structured illumination [27] and line-confocal detection [13] have been proven to be appropriate methods for this purpose. The latter is of particular interest because it performs well in strongly scattering samples and can be easily implemented by means of rolling shutters in CMOS cameras [14]. However, not all out-of-focus fluorescent photons are blocked by the confocal slit. Even if the out-of-focus light passing the slit is negligible, the fluorescent photons blocked by the slit are not imaged, but contribute to photobleaching. Therefore, the power in the ring system has to be minimized in order to optimize the performance of the microscope.

However, with the following approach, the Bessel beam’s bleaching problem can be strongly reduced: For a predefined NA of the Bessel beam, the power in the ring system can be minimized by tailoring the light-sheet, i.e. restricting the length of the beam (at full width half maximum, FWHM) to the length of the object. In a tailored light-sheet, the length and the focus position of the illumination beam is adapted to different positions within the object, while the beam is scanned through the focal plane.

Such light-sheet tailoring has been demonstrated for Gaussian beams by Chmielewski et al. [28], where the Rayleigh length and the axial beam center were adapted by means of two tunable lenses. However, the low response time of the lenses limited the framerate to 5 Hz. Fast scanning of a focused Gaussian beam in axial direction [29] combined with simultaneous modulation of the beam power would be another possibility to form tailored light-sheets with high framerates, but has not been demonstrated yet.

In this work, we show that tailored LSM with Bessel beams strongly enhances the contrast and reduces bleaching. We explain a principle where amplitude shaping of Bessel beams can be realized at more than 10 kHz and how this enables tailored LSM in real time at framerates of 50 Hz and hence with 200 different beams per frame.

2. Basic principle

A schematic drawing of the tailored light-sheet microscope is illustrated in Fig. 1 for both the pre-scan mode and the final adaptive, fluorescence mode. A spatial light modulator (SLM; Holoeye; Pluto-NIR-II) is used for beam shaping. The hologram is calculated such that a divergent beam is reflected from the SLM, or in other words, a virtual Bessel beam forms at the backside of the SLM [30]. The lens L1 focuses the annular spectrum of the Bessel beam onto the scan mirror (SM). The mirror is located in a plane conjugated to the back focal plane (BFP) of the illumination objective (IO; Mitutoyo, M Plan Apo NIR 20x), such that a tilt of the mirror results in a scan of the beam through the focal plane of the detection objective (DO; Zeiss, W Plan-Apochromat 20x/1.0). For simplification, an additional 4f-System between scan mirror and BFP is not depicted. A detailed description of the setup can be found in previous publications [30].

 

Fig. 1 Setup sketch for a light-sheet microscope using length-adapted Bessel beams. Here, a SLM is used for beam formation. The scan mirror (SM) placed in a plane conjugated to the BFP of the illumination objective (IO) scans the phase shaped beam through the focal plane of the detection objective (DO). (a) In a pre-scan, the shape of the object is estimated by using constant, long Bessel beams and scattered laser light imaged to a camera. (b) Different amplitude modulations of the holograms on the SLM adapt the focal positions and the beam lengths to the shape of the object (depicted in red) varying at each scan position. The DO and a tube lens (not shown) image the fluorescent light onto the camera (Hamamatsu, OrcaFlash 4.0 v2), equipped with a rolling shutter (slit) to enable line-confocal detection.

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The shape of the object (sketched in red in Fig. 1) or the sufficiently well illuminated part of the object does not fill the whole field of view of the detection system. Therefore, at most scan positions the beam is longer than necessary, which is a typical case in LSM. Adapting the beam to the object shape at each position x via the SLM minimizes the average beam length Δz(x) and thus enhances contrast and reduces bleaching. The shape of the object and therefore the ideal light-sheet in the observed xy-plane $S\left(x,z\right)$ can be described by the binary function

Here z₀(x) describes the x-dependent center of the object in z-direction (see dashed line in Fig. 1) and Δz(x) represents the x-dependent extent of the object in z-direction. The shape of the ideal light-sheet is determined from the laser light scattered by about 90° to the camera, producing a scattered light image of the contour of the object.

Figure 2 shows the basic principle of light-sheet tailoring: In the first step, a scattered light image is captured within less than 0.1s in normal light-sheet mode but without the emission filter, usually blocking the laser light. The coherent image Fig. 2(a) requires very low illumination power, so that the object is not stressed by the additional illumination and bleaching can be neglected.

 

Fig. 2 Basic principle of tailored LSM. a) The object shape and thus the shape of the ideal light-sheet (white area in (b)) is estimated based on an image of purely scattered laser light as shown in (a). c) For the rectangular reference light-sheet (LS) shown in blue a single beam is predefined and not adapted throughout the scan. The corresponding beam forming hologram is depicted on the right. The enlarged view shows the wrapped conical phase and the random binary amplitude modulation [30]. d) In tailored LSM, the axial position and the beam length is adapted to the object shape (length adapted LS in blue). The corresponding holograms, which change with the varying object shape, are depicted on the right of the respective beam. By beam adaptation, the mean beam length and in consequence the power in the ring system of the Bessel beam is minimized.

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In conventional LSM, the object point closest to the illumination objective z₁ and the point furthest away from the objective z₂ defines length and focal position of the ideal beam Fig. 2(b). The shape of the beam is not changed throughout the scan. That way the standard rectangular light-sheet forms, which we call reference light-sheet Fig. 2(c). In tailored LSM, the shape of the ideal light-sheet is reproduced by updating the beam forming hologram after a certain scan distance Fig. 2(d)).

A conical phase $\varphi (x,y)=k_0\cdot \text{NA}\cdot \sqrt{x^2+y^2}$ is required in order to form a Bessel beam, where the phase slope k₀∙NA defines the NA of the beam (at wave number k₀ = 2π/λ₀ with vacuum wavelength λ₀). Remarkably, the phase of the hologram is independent from the axial beam profile, i.e. the phase and the NA are not changed during the measurement. To adapt the beam length Δz(x) and axial beam position z₀(x), only an amplitude modulation is required. Figure 3(a) shows how amplitude and phase of the hologram displayed on the SLM define the beam shape.

 

Fig. 3 Bessel beam length, ring energy and signal-to-background ratio. A) Principle of beam formation. Only the amplitude of the hologram defines the length Δz and center position z0 of the beam. The conical phase ϕ(r) (depicted by the white lines) defines the NA of the beam and is identical for all 3 beams. b) Calculated relative energy contribution q_i of the individual Bessel rings at the image formation for 4 different beam lengths Δz. q₀ represents the main maximum, q_n the n-th ring in the ring system. d_s is the confocal slit width. c) The signal–to-background ratio (SBR) decreases with the beam length Δz for conventional (d_s → ∞) and confocal (d_s = 0.1 µm) detection.

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We assume the typical case that the distance z₀ between axial beam center position and hologram position is larger than the beam length Δz. Then the transmission function T(x,y) and the amplitude A(x,y) of the hologram have an annular shape [30,31]:

With $r=\sqrt{x^2+y^2}$ and circ(r-r₀) = 1 for r ≤ r₀ and = 0 otherwise. The center diameter R₀ of the ring defines the axial beam position according to $z_0=n\cdot R_0/\text{NA}$ with n being the refractive index. The beam length is given by the width ΔR of the ring according to $\Delta z=n\cdot \Delta R/\text{NA}$. The magnification of the illumination system has been set to M_ill = 1 in order to keep things simple. To avoid oscillations of the axial beam profile a low-pass filtering of the amplitude A(x,y) is required.

The SLM used in the current setup is a pure phase shifting element, although we used an algorithm to modulate the amplitude as well by transforming the continuous amplitude modulation into a binary pattern [30]. Therefore, in order to modulate beam length and axial beam position, a dynamic binary amplitude modulation in combination with a static phase modulation can replace the often slow SLMs. Hence, an axicon can be used as static phase element and a digital micromirror device (DMD) for dynamic binary amplitude modulation. The framerate of the currently used SLM is limited to 60 Hz, which is not enough to enable real time light-sheet adaptation. In contrast, DMDs feature framerates of more than 10 kHz [32–34]. Thus, for a field of view of 500 µm and a framerate of 50 Hz, totally 200 beams can be adapted laterally in steps of 2.5 µm within only 20 ms. This step size is small enough to sample all relevant object shapes. Also binary phase SLMs have been used for fast Bessel beam position control [35]. In this study, beams were rapidly shifted axially to enabled tiled light-sheets of rectangular shape, hence, light-sheets were not adapted to the shape of the object.

Figure 1 shows a confocal detection based on the rolling shutter technique, which requires a minimum scanning speed (minimum speed of rolling shutter). For the proof of principle measurements presented in this work, a DMD has not been available, so that beam adaptation could not be realized at the scanning speed required for rolling shutter based confocal detection. To circumvent this problem, individual images were captured automated at each lateral scan position. The final image was calculated by multiplying the individual images with a mask representing the confocal slit and subsequently summing up all images [13]. To enable a fair comparison of imaging with tailored and untailored light-sheets, without being influenced by sample drift or bleaching, an image with and without beam adaption was recorded at each scan position before the beam was moved to the next position. This workaround performed well for our proof-of-principle experiments, clearly demonstrating the novel beam shaping principle, but is much too slow for most applications. Again, we want to point out that real time light-sheet tailoring in combination with rolling shutter based confocal detection is possible by implementing a DMD as fast 2D amplitude mask. DMDs with the same format and pixel size as the used SLM are commercially available and enable beam adaption with more than 10 kHz. In addition, DMDs provide onboard memories, which can be used as a library for different beam forming amplitude patterns.

As explained above, the power in the ring system of the Bessel beam increases with the beam length. Figure 3(b) illustrates the influence q_i of the individual rings onto the formation of the image in dependency of the confocal slit width ds and the beam length Δz. (See the method section for more information about the calculation of q_i.) The distribution for an infinite slit width $d_s\to \infty$ is identical to the power distribution in the beam. The bar chart shows that the proportion of photons in the main maximum - with FWHM of 0.88 µm at NA = 0.2 - is proportional to 1/Δz. Assuming that only photons emitted in the main maximum contain useful image information, the total beam power must be proportional to the beam length, if the useful signal level should remain constant. However, with increasing beam length the total beam power and therefore the amount of bleaching increase. In other words, a reduction in beam length is proportional to a reduction in bleaching.

The bar chart in Fig. 3(b) shows the fraction q₀ of photons that form a useful light-sheet image i.e. photons excited in the main maximum. This fraction q0(Δz) depends on the beam length Δz and, in the case of confocal detection, on the slit width d_s. For a slit width of d_s = 1 µm, one finds q₀ = 54% for Δz = 200 µm and q₀ = 42% for Δz = 600 µm. Assuming that only the photons excited in the main maximum contribute to the useful signal, whereas all photons from the Bessel rings form the image background, the signal-to-background ratio is given by

Therefore, the SBR increases by 62% if the beam length is reduced from 600 µm to 200 µm. Figure 3(c) shows the decay of SBR(Δz) with beam length Δz for conventional and confocal detection mode. In conventional mode, the relative improvement achieved by keeping the illumination beam short is more pronounced than in the confocal case. Nevertheless, the absolute SBR is always higher if confocal detection is applied.

3. Results

3.1 Increasing contrast with adaptive light sheets

To demonstrate the contrast enhancement capabilities of tailored LSM, we imaged an Arabidopsis root tip with (LTi6b:eGFP) membrane labeling and a fixed spherical cell cluster of T47D breast cancer cells stained with propidium iodide DNA labeling. The ideal light-sheet and the reference light-sheet – without object adaptation – used for the Arabidopsis root are presented in Fig. 2(c),(d). The NA of the illumination beam has been set to 0.13, resulting in a FWHM of the first maximum of 1.36 µm. Figure 4 shows a comparison of the corresponding fluorescent images using the reference light-sheet and a tailored light-sheet (both using line-confocal detection). The magnified image sections A and B underline the improvement in image contrast. To determine the contrast enhancement quantitatively, the Fourier transformations $P_{ref}\left(k_r,\theta \right)$ and $P_{tay}\left(k_r,\theta \right)$of the reference image p_ref(r,φ) and the tailored image p_tay(r,φ) were calculated and averaged over the polar angle: ${\overline{P}}_{xx}\left(k_r\right)={\displaystyle {\int }_0^{2\pi }\left|P_{xx}\left(k_r,\theta \right)\right|}d\theta$. For better comparability, both spectra have been normalized to the same noise level and an identical DC component. The ratios of the tailored and reference spectra, $P_{tay}\left(k_r\right)/P_{ref}\left(k_r\right)$ are plotted separately for the upper left rectangular area (Region 1) and lower right area (Region 2) of the root image. Because all spectra have been normalized to the same DC component, the ratio $P_{tay}\left(k_r\right)/P_{ref}\left(k_r\right)=1$for k_r = 0. It can be seen that the amount of spatial frequencies above DC are enhanced up to 50% by light-sheet tailoring, i.e. $P_{tay}\left(k_r\right)/P_{ref}\left(k_r\right)\approx 1.5$. This corresponds to a background reduction of 33%. The effect reduces with increasing frequency because noise becomes more dominant in this range. The contrast improvement is more pronounced in the lower right part of the image. This can be explained by the shorter beam length which was applied in this area for tailored imaging. This effect is especially pronounced in the conventional detection mode. Although contrast improvement is better for the conventional detection, the pure image contrast with confocal detection is still much better than with conventional detection - with and without adaptive Bessel beams.

 

Fig. 4 Fluorescent images of an Arabidopsis root tip without (a) and with (b) light-sheet tailoring. The images were recorded with confocal detection at a slit width of d_s = 1 µm. The beam length was adapted to the object, the illumination NA was constantly 0.13 (FWHM = 1.36 µm). The NA of the detection objective was 1.0. (c) The magnified ROIs A, A’, B, B’ and (d) the normalized image spectra as a function of radial spatial frequency (bottom right) illustrate the contrast improvement by light-sheet tailoring. The increased image contrast for different spatial frequencies are shown also for the conventional detection mode with d_s = 50 µm.

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3.2 Adapting the beam length to the penetration depth

The line-confocal light-sheet fluorescence images of the cell cluster are shown in Fig. 5(a),(b). The cells had been in culture for 28 days, growing to a diameter of roughly 500 µm of the cell cluster. In consequence, the penetration depth of the illumination beam is limited due to scattering and absorption.

 

Fig. 5 Images of a spherical cell cluster with and without light-sheet tailoring. (a) The purple line outlines the rectangular reference light-sheet and (b) the tailored light-sheet, which is defined by the penetration depth of the illumination beam. The images were recorded with confocal detection at a slit width of d_s = 1 µm. The illumination NA was 0.13 (FWHM = 1.36 µm), the detection NA was 1.0. (c) The ratio of the averaged image spectra as a function of radial frequencies shows a contrast improvement of 70% (background reduction of 41%) by light-sheet tailoring. d) The fluorescence line scan marked by the white dashed line also shows a contrast improvement.

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In such a case, the ideal light-sheet is not only defined by the object shape, but also by the penetration depth. The purple lines outline the reference light-sheet and the ideal light-sheet for the investigated cell cluster. Again, a magnified region of interest reveals the contrast improvement. This finding is quantified in Fig. 5(c) by the ratio of the averaged image spectra $P_{tay}\left(k_r\right)/P_{ref}\left(k_r\right)$, which increased by up to 70% and the line scans (Fig. 5(d)). The position of the line scan is marked by the dashed white line in the magnified region of interest. In some areas of the image the intensity with the tailored light-sheet is lower than with the reference light-sheet, which is caused by the shape of the illumination beam. By definition the maximum intensity of the beam profile at the beginning and end of the beam should be half the maximum intensity of the beam center. For this reason the intensity at the border of the ideal-light sheet is lower for the tailored light-sheet than for the reference light-sheet.

3.3 Reduction of bleaching with adaptive light sheets

The influence of the beam length on fluorophore bleaching was investigated by the example of a zebrafish. The fish labeled with Alexa 555 was illuminated with two different Bessel beams at a wavelength of 561 nm. The beams have the same NA = 0.13, but differ in length because of a different amplitude filter on the SLM. The longer beam with a length of Δz = 500 µm was used to bleach the tail of the zebrafish at position x₁. The shorter beam with Δz = 200 µm exposed the fish at position x₂. The fluorescent light excited along the two beams is shown in Fig. 6(a) left. Because the width of the tail is smaller than 200 µm, the different beam length can only be recognized by the width of the beam. The longer beam is surrounded by a more pronounced ring system and thus appears broader. The beam power - proportional to the beam length - has been adapted in a way that the maximum intensity at the beam center is the same for both beams.

 

Fig. 6 Fluorophore bleaching of longer and shorter Bessel beams. (a) Left: two Bessel beams with the same NA, but different lengths illuminate and bleach the fluorophores of a zebrafish tail at two different x-positions. Right: Images of the zebrafish tail before bleaching (t_b = 0) and after bleaching (t_b = 4 s and t_b = 8 s). (b) A kymograph, i.e. a z-projection of the images reveals that bleaching by the longer beam is much stronger. (c) The value N_p describes the loss of fluorescent power in a certain x-interval. Thus, N_p is proportional to the percentage of bleached photons.

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For the investigation of the different bleaching behavior of both beams the following experiment was carried out. First, a light-sheet image of the zebrafish tail has been captured. Subsequently, the fish was illuminated continuously for 2 seconds by the longer beam followed by an illumination for again 2 seconds with the shorter beam. Such bleaching cycles have been repeated until no change of the fluorescent signal could be recognized. Figure 6(a) shows the light-sheet image before any bleaching occurs (t_b = 0 s). The fluorescence is equally distributed over the tail. After two bleaching cycles or a bleaching time of tb = 4 s respectively, the slight bleaching can be recognized at the position of the shorter beam x₂, while a significant drop of the fluorescent signal shows up at position x₁. For a better illustration of the bleaching process, the images were integrated in z-direction and normalized to 1 at bleaching time t_b = 0: $\overline{I}\left(x,t_b\right)={\displaystyle \int I\left(x,z,t_b\right)}dz/{\displaystyle \int I\left(x,z,t_b=0\right)}dz$.

The kymograph $\overline{I}\left(x,t_b\right)$in Fig. 6(b) shows that the region bleached by the longer beam darkens much faster than the region bleached by the shorter beam. In a next step the intensity loss caused by the respective beam has been integrated over an x-interval with width 2b = 80 µm:

The value N_p is proportional to the percentage of bleached photons and is plotted in dependency of the bleaching time t_b in Fig. 6(c). One can see that the longer beam bleaches twice as many fluorophores as the shorter beam.

4. Discussion

By imaging an Arabidopsis root tip and a spherical cell cluster it has been shown that the contrast can be increased over a reasonable range of spatial frequencies by 50% and 70%, respectively (Fig. 4). Theoretical considerations based on Eq. (3) predicted a contrast (SBR) improvement of 62%, for a reduction in beam length from 600 µm to 200 µm (Fig. 3), which is close to the contrast improvement of 50% and 70%. In the case of the root tip, the reference beam (no adaptation) had a length of 570 µm, corresponding to the overall extent of the object. However, when the much shorter tailored beam was varied between 100 and 150 µm, the contrast improvement of 50% was good, but slightly lower than expected. This can be explained by the fact that the composition of the image from the individual ring contributions, shown in Fig. 3(b), was calculated for the beam center. The detrimental contribution of the ring system increases with the distance to the beam center. This effect is stronger for shorter beams and leads to a reduced contrast improvement compared to the theoretical prediction.

Whether a clever positioning of the root tip can make the tailoring of the light-sheet obsolete remains open for discussion. However, if for example the growth process of the root tip is of particular interest, gravity will predefine an upright position of the plant and thus the roots are usually not parallel to the axes of the microscope.

Furthermore, most light-sheet microscopes allow a rotation only around the x-axis (perpendicular to both optical axes). By rotating the root by 90° around x, light-sheet tailoring becomes unnecessary, but at the same time the size of the object in detection direction would increase and thus also the number of frames per 3D-stack.

According to the bar chart shown in Fig. 3(b), it was expected that bleaching is proportional to the beam length. The experimental data presented in Fig. 6 shows that the percentage of bleached fluorophores increases by a factor of 2, if the beam length is scaled up by a factor of 2.5. This can be explained by the fact that the light-sheet images have been captured with line-confocal detection, where the influence of the bleached areas far away from the focal plane is reduced. This is more relevant for longer beams, which are significantly broader than the shorter beam. Nevertheless, we could demonstrate a tremendous effect of the beam length on fluorophore bleaching. Therefore, we expect that the possible acquisition time of biological specimen sensitive to fluorophore bleaching can be doubled by light-sheet tailoring. This nearly compensates the main disadvantage of Bessel beams relative to Gaussian beams, which is enhanced bleaching, but maintains the strong advantages of Bessel beam illumination such as higher axial resolution and more homogeneous illumination.

5. Conclusion and outlook

We presented a concept for object adapted light-sheet tailoring. Adapting the Bessel beams lengths and positions to the space variant extent of the biological object result in a significant improvement in contrast and reduced photo bleaching. The image improvement depends on the shape and orientation of the object, which was estimated by a pre-scan using low-power scattered laser light. Light-sheet tailoring can be added to any system using Bessel beams, which are generated by either a static phase axicon or spatial light modulators, and this without any drawbacks or compromises. Especially photobleaching is expected to be reduced by light-sheet tailoring, although this effect has to be validated with different specimen in long term imaging experiments in the future.

Due to the characteristics of Bessel beams, light-sheet tailoring requires only binary amplitude modulation, which is also possible by phase-only SLMs. Due to the low frame rate of conventional SLMs, the proof of principle measurements presented in this article have been captured with a very limited frame rate. However, because only binary amplitude modulation is required, the phase-only SLM can be replaced by a fast DMDs or binary SLM. These devices feature frame rates, which enable tailored light-sheet microscopy at the maximum imaging speed of modern sCMOS cameras. With pure amplitude modulation, it is not possible to adapt Gaussian beams, which require a phase modulation in order to optimize the focal position and the depth of field.

Beside the automated object adapted light-sheet tailoring, the shape of the light sheet can also be defined by the user. In this way, parts of the object can be excluded from illumination. It is also possible to create light-sheets with user defined intensity distributions. In scanned LSM this is easily achieved in scanning direction by modulating the illumination intensity. In addition, the presented technique allows adaptation along the illumination direction. This enables the compensation of a varying fluorophore density as well as the correction of the intensity drop due to absorption of the illumination beam.

In summary, the simple and straightforward automation of tailored light-sheets enables this concept promising to become a standard feature in LSM with Bessel beams.

6. Method

Simulating the contribution q_i of the individual Bessel rings onto the image formation a) Calculate the 3D intensity distribution of Bessel beams with different lengths by the angular spectrum wave propagation method. b) Separate the beam into its rings. c) Separate convolution of the Beam rings with the 3D-Detection-PSF in order to get the images of the individual beam rings for the case the object is a homogeneous distribution of fluorophores. d) Cut the focal plane out of the 3D images. e) Calculate the ratio q_i of the power inside a slit of width ds for a certain ring i and the total power in the image of all rings.