## Abstract

One of main challenges in light-sheet microscopy is to design the light-sheet as extended and thin as possible - extended to cover a large field of view, thin to optimize resolution and contrast. However, a decrease of the beam’s waist also decreases the illumination beam’s depth of field. Here, we introduce a new kind of beam that we call sectioned Bessel beam. These beams can be generated by blocking opposite sections of the beam’s angular spectrum. In combination with confocal-line detection the optical sectioning performance of the light-sheet can be decoupled from the depth of field of the illumination beam. By simulations and experiments we demonstrate that these beams exhibit self-reconstruction capabilities and penetration depths into thick scattering media equal to those of conventional Bessel beams. We applied sectioned Bessel beams to illuminate tumor multicellular spheroids and prove the increase in contrast. Sectioned Bessel beams turn out to be highly advantageous for the investigation of large strongly scattering samples in a light-sheet microscope.

© 2013 OSA

## 1. Introduction

Light-sheet microscopy provides true optical sectioning by sample illumination with a thin light-sheet. The technique is able to deliver high-contrast images of large samples at high speeds with low photo-damage and bleaching. Conventionally, samples are illuminated by a static light-sheet formed by a strongly elliptical beam from one [1] or two sides [2]. Alternatively, Gaussian beams [3] or Bessel beams [4–6] are laterally scanned in the plane of focus. However, for all illumination techniques presented so far, the thickness of the illuminated volume increases with the size of the image along the illumination optical axis. Therefore, optical sectioning and image contrast decrease for large samples.

Superior contrast and improved resolution were achieved for Bessel beam illumination by employing a confocal-line detection scheme [7]. The principle of confocal-line detection light-sheet microscopy consists in acquiring the image line-wise at the position of the illumination beam while it is scanned across the sample. The method is also able to increase contrast for Gaussian beam illumination [8, 9], if the beams exhibits sufficient propagation stability along a line.

Here, we introduce a new kind of beam that we call sectioned Bessel beam (SBB) that can be generated by blocking opposite sections of the beam’s angular spectrum. Sectioned Bessel beams are similar to Mathieu beams, a solution to the Helmholtz-equation in parabolic coordinates. But even though Mathieu beams have been investigated for a long time [10, 11], no study on their propagation performance in strongly scattering media has been reported so far. First, we give analytical descriptions for shape and dimensions of the SBB. We further present results from computer simulations with the Beam Propagation Method (BPM) and analyze the penetration depth into scattering media. Using simulation data, we demonstrate in section 6 the influence of the section angle of the SBB’s angular spectrum on the optical sectioning capability of the light-sheet. Moreover, we show that SBBs in combination with confocal-line detection allows to decouple the optical sectioning from the depth of the field of the illumination beam and thereby the object size. Finally, we analyze the performance of SBBs in strongly scattering tumor multicellular spheroids with regard to penetration depth and image contrast.

## 2. Sectioned Bessel beams - shapes and dimensions

A typical (zero-order) Bessel beam used in light-sheet microscopy is characterized by its angular spectrum *Ẽ _{B}*, which consists of a thin ring of finite thickness. It can be described by${\tilde{E}}_{B}({k}_{x},{k}_{y})={\tilde{E}}_{0}\left(\Theta ({k}_{0}{\text{NA}}_{B}-{k}_{r})-\Theta (\sqrt{\epsilon}\text{\hspace{0.17em}}{k}_{0}{\text{NA}}_{B}-{k}_{r})\right)$ with ${k}_{r}={\left({k}_{x}^{2}+{k}_{y}^{2}\right)}^{1/2}$. Here Θ(k) denotes the Heaviside step function and NA

_{B}

*= n⋅*sin

*α*describes the maximum focusing angle of the Bessel beam. ε < 1 is the ring width parameter, i.e. the area ratio of the inner to the outer disc [4]. The outer radius of the annular spectrum is

*k*

_{0}NA

_{B}, the ring thickness is

*k*

_{0}NA⋅(1-√ε). For ε = 0, the back aperture is homogeneously illuminated and a conventional beam with a limited depth of field $dz=\lambda /(n-n\mathrm{cos}\alpha )$ is created.

Other than Bessel beams with a complete ring-shaped angular spectrum, the spectrum of a sectioned Bessel beam consists of two opposed sections of the ring - each spanning over an angle β as pointed out in Fig. 1(a) in the *k _{x}k_{y}*-plane. The red-shaded areas are projections of the Ewald-sphere onto the three orthogonal planes in k-space. In the

*k*-plane the angular spectrum representation

_{x}k_{y}*Ẽ*(

*k*,

_{x}*k*) exhibits a point symmetry with respect to the optical z-axis. Mathematically, the spectrum can be sufficiently well described by

_{y}*x*/

*b*) = 1 if |

*x*|≤

*b*/2 and 0 otherwise. β = 180° results in a conventional Bessel ring spectrum, since the rect-function does not cut-off any frequencies.

Correspondingly, the field of a Bessel beam in real space can be obtained by an inverse Fourier transform:

*E*

_{SeB}(

*x,y*)|

^{2}, |

*E*

_{SeB}(

*x,z*)|

^{2}and |

*E*

_{SeB}(y

*,z*)|

^{2}are shown in Fig. 1(b). Here and throughout this study

*x*is the beam scanning direction parallel to the light-sheet,

*y*is the direction of the detection optical axis perpendicular to the light-sheet.

It can be seen that the Bessel rings are suppressed in the detection *y*-direction because of the convolution with the sinc-function, which has a slightly different frequency than the *J*_{1}-functions (Eq. (2)) along *y*. This has important implications for the application in light-sheet microscopy as will be shown below.

From Eq. (1) and by inspecting the corresponding Fig. 1(a), it is straight forward to estimate the dimensions of the main lobe of the sectioned Bessel beam. These are defined by the inverse of the extents *dk _{x}*,

*dk*and

_{y}*dk*, as projections of the spectrum ${\tilde{E}}_{\text{SeB}}({k}_{x},{k}_{y},{k}_{z})$ onto the axes and can be described by the adjustable parameters ε, β and NA

_{z}_{B}. The outer-diameter of the ring-spectrum is 2

*k*

_{0}⋅NA

_{B}and is defined by the NA of the focusing lens, NA

_{B}. The inner diameter is 2√ε⋅

*k*

_{0}⋅NA

_{B}. The extent of the angular spectrum ${\tilde{E}}_{\text{SeB}}({k}_{x},{k}_{y})$ in

*k*-direction is solely determined by the section’s angular extent, β, such that

_{y}*dk*= 2

_{y}*k*

_{0}⋅NA

*⋅sin(β/2), i.e. the width of the rect-function. The ring thickness determines the axial extent of the spectrum*

_{B}*dk*, which scales with the ring parameter ε such that

_{z}*dk*=

_{z}*k*

_{0}⋅(cosα - ε⋅cosα) ≈2

*k*

_{0}⋅NA

_{B}^{2}⋅(1-ε) in the paraxial approximation.

As shown in Fig. 1(a) the spectrum in *k _{x}*-direction consists of two fragments of width

*dk*in an outer distance of Δ

_{x}*k*= 2

_{x}*k*

_{0}⋅NA

_{B}to each other. The frequency Δ

*k*defines the radial ring spacing of a Bessel beam. After projection in the pupil plane (angular spectrum), the (red) ring area on the Ewald sphere consists of the ring thickness (1-√ε) and the (1-cos(β/2)) part (see distance Δ

_{x}*k*indicated in green in Fig. 1(a)), such that

*dk*=

_{x}*k*

_{0}⋅NA

*⋅(1-√ε) + √ε⋅*

_{B}*k*

_{0}⋅NA

*⋅(1-cos(β/2)) =*

_{B}*k*

_{0}⋅NA

*⋅(1-√ε⋅cos(β/2)).*

_{B}*dk*defines the width of the beam envelope of the sectioned Bessel beam in scan direction as shown in Fig. 1(b).

_{x}From the uncertainty relation *dj*⋅*dk _{j}* = 2π (

*j = x,y,z*) we obtain the following extents of a sectioned Bessel beam with section angle β:

*dx*,

*dy*and

*dz*represent effective widths and correspond to real space standard deviations obtained by the inverse of the standard deviations in

*k*-space [12]. The full width (root distance)

*dx*in scan direction describes the envelope of the modulated beam, but does not describe a Bessel beam with

*β*= 180°. Whereas

*dz*is independent of β, the dependency of the beam widths

*dx*and

*dy*on the section angle β is displayed in Fig. 2(a) (with ε = 0.8 and NA

_{B}= 0.4). It can be seen that

*dx*(β) and

*dy*(β) become minimal for β approaching 180°, which corresponds to a conventional Bessel beam with finite ring width. However, for a reasonable range of section angles β > 60°, it turns out that the width

*dy*in detection direction hardly changes with β, whereas

*dx*falls of strongly. In other words, the light sheet thickness

*dy*is nearly independent of β for β > 60°.

As shown in Fig. 2(b), a Bessel beam’s depth of field *dz* can be steered efficiently by varying the ring parameter ε = 0.7…0.95. This has a small effect on the lateral width *dx* and no effect on *dy*. By plotting *dy*(NA) against *dz*(NA) for the Gaussian beam (ε = 0 and β = 180°) as shown in Fig. 2(c), one finds that for a decreasing NA or increasing depth of field *dz*(NA) the width *dy*(NA) increases. For the conventional Bessel beam (β = 180°) and the sectioned Bessel beam (β = 100°), the widths *dy*(ε) remain constant when increasing *dz*(ε) via the ring parameter ε. As we will show below, this situation changes when the detection point-spread function PSF_{det}(*x*,*y*,*z*) is considered.

Note that these measures do not account for the total fluorescence excited by the beams. These beam extents do not represent the optical sectioning capability, i.e. how well the beams are suited in order to detect fluorescence only from a thin region around the focal plane of the detection objective lens. Data on the optical sectioning performance will be shown in sections 5 & 6.

## 3. Measures for directional propagation stability

So far, only beam widths for free-space propagation have been considered. Analysis of the propagation behavior of sectioned Bessel beams in inhomogeneous media yields a quantitative measure for their self-reconstruction ability. First, we pursued a numerical beam propagation approach.

In the following, a measure for directional propagation stability is introduced. To assess how strongly the beams are scattered or deflected by obstacles, we measure the power that the beam carries up to a certain radius around its propagation axis:

*I*(

*x*,

*y*,

*z*) = |

*E*

_{SeB}(

*x,y,z*) +

*E*

_{sca}(

*x,y,z*)|

^{2}is the interference of the unscattered field

*E*

_{SeB}(Eq. (1)) and the manifold scattered field

*E*

_{sca}. By normalizing the power of the beam propagating through a scattering sample

*P*

_{scat}to the power

*P*

_{ideal}of the corresponding unscattered beam one obtains

*Q*gives the relative power that a beam is able to maintain within a small radius

*R*around its axis in the presence of scatterers.

*Q*It is robust against natural beam spreading due to diffraction in homogeneous space and comparable for beams which differ in irradiance and total power. The power ratio averaged over Δ

*z*along the propagation axis

^{7}:1 in comparison to the full volume data.

A measure for directional propagation stability can be used to answer the following question: Does the beam follow its initial direction even in the presence of phase perturbations induced by scatterers? Or, put in a more quantitative way: How much energy is taken out of the beam’s central region (radius *R*) relative to the ideal, unscattered beam as a function of propagation distance *z*? This measure is relevant to microscopy: Good images with homogeneous illumination require that this power is independent of the perturbation of the beam. The change of the beam’s on-axis power at different positions in a sample that arises from different scattering of the beam is a major source of image artifacts. Its strength can be measured by the standard deviation of the position dependence of the on-axis power [5].

## 4. Simulation results for directional propagation stability

At first, we numerically investigated the propagation of coherent beams through scattering samples using the Beam Propagation Method (BPM) [13, 14]. The standard Gaussian beam focused by NA = 0.08 is compared to two Bessel beams: one with a low NA (NA = 0.2, ε = 0.74) and one with a high NA (NA = 0.4, ε = 0.94). Additionally, a sectioned Bessel beam with a high NA (NA = 0.4, ε = 0.94) and an azimuthal width of the section of β = 88° was analyzed. All beams exhibit equal depth of field of *dz*≈100µm. Cross-sections of the beams are shown in Fig. 3.

The scattering medium is represented by a distribution of silica spheres with refractive index of *n*_{scat} = 1.41 embedded in an aqueous gel with *n*_{med} = 1.33. A small imaginary refractive index (*n _{i}* = 0.0004) is included to account for the beam energy that is lost due to backscattered light, which the BPM cannot account for otherwise. The simulations were performed for different parameters of the scattering sample, i.e. volume concentrations (ρ = 6%, ρ = 12%) and size (

*d*= 2µm,

*d*= 4µm). To obtain representative results, the beams were propagated at 9 different (

*x*

_{i},

*y*

_{i}) positions for 15 different arrangements of the spheres expressed by

*n*(

_{j}*x*,

*y*,

*z*). The stability ratio $\overline{Q}(z)$ was then computed as the average over all 9 beam positions in all 15 samples.

The results are shown in Fig. 4. It can be seen that $\overline{Q}(z)$ is similar for the Gaussian beam and the low-NA Bessel beam. For both beams the values of $\overline{Q}(z)$ for large *z* are inferior to those for the high-NA Bessel beam and surprisingly also the sectioned Bessel beam with a high NA. This result means that the relative power of the beam within a radius *R* around the propagation axis is higher for (sectioned) Bessel beams if the NA of the beam is high. The penetration depth is higher for larger spheres for all illumination beams. These obstacles scatter more light in forward direction, i.e. they remove less light from the beams. This result qualitatively agrees with a rough estimate using the reduced scattering coefficient µ’_{scat} = (1-*g*)⋅µ_{scat} where *g* is the size-dependent scattering anisotropy factor (for λ = 488nm: *g*_{2µm} = 0.96, *g*_{4µm} = 0.99).

The data obtained was analyzed in more detail. Figure 5 shows the average power ratio along the simulated volume according to Eq. (6) in dependence of the beam position *i* (lateral axis) and the volume *j* (vertical axis). The magnitude of ${\overline{Q}}_{i,j}$ is coded by the color of the spot for each of the beam positions *i* = 1...9 in each of the volumes *j* = 1...15 corresponding to 135 simulations for each beam type. While white represents ${\overline{Q}}_{i,j}$ = 1, meaning that the power is approximately equal in the scattered and unscattered case, green and blue spots indicate weak intensity and a low power of the beam in the area around the propagation axis. Note that the relative on-axis power decreases monotonously, which is a prerequisite for the conclusions drawn from the values of ${\overline{Q}}_{i,j}$. A large difference between the results for a Gaussian beam and the various Bessel beams can be observed. The variance of the values ${\overline{Q}}_{i,j}$ is high for the Gaussian beam while it is small for the Bessel beam. This finding indicates the stronger susceptibility of the Gaussian beam to beam deflections and scattering.

It is interesting to inspect the intensity distribution of the beams for two extreme situations, i.e. position *i* = 1 in volume *j* = 10 where ${\overline{Q}}_{i,j}$ is exceptionally high for the Gaussian beam and position *i* = 9 in volume *j* = 4 where ${\overline{Q}}_{i,j}$ for the Gaussian beam is very low (marked by arrows in Fig. 5). In the first case, the Gaussian beam propagates through a region practically without spheres and therefore remains almost unperturbed, whereas in the second case obstacles close to the beam axis perturb the beam strongly and deflect it from its initial propagation direction. In contrast, the Bessel beam and the sectioned Bessel beam generated at a high NA = 0.4 exhibit a very weak perturbation in both cases that mainly affects the ring system. Most importantly, the main lobe is neither deflected nor strongly distorted. It can be seen in Fig. 3 that for *i* = 1 & *j* = 4, i.e. an average value of ${\overline{Q}}_{i,j}$, the central peak of both beams is quite well maintained of the whole distance. The main lobe of the Bessel beam is much more confined and is well separated from the ring system, whereas the Gaussian beam is blurred over a larger area.

From the simulation results shown in Figs. 4 and 5 the following conclusions can be drawn. Gaussian beams show penetration depths and directional propagation stabilities that are inferior to those of high-NA Bessel beams, but not to those of low-NA Bessel beams. Remarkably, high-NA sectioned Bessel beams are equally robust against scattering as Bessel beams with the same NA. Moreover, Gaussian beams are much more sensitive to local perturbations by scatterers so that the average on-axis intensity is more position dependent. This difference between Gaussian and Bessel beam is partly due to the larger cross-section of the Bessel beams: more scatterers affect the beam, but the impact of each single scatterer is much lower.

## 5. Theoretical considerations on optical sectioning

The profile of a sectioned Bessel beam is characterized by the fact that the central lobe of the beam is mainly fed by self-healing photons from the sides along the scan axis *x*. If sectioned Bessel beams are used to illuminate a thin line and fluorescence along this line is detected using the confocal-line detection scheme [7] it is expected that less background will be recorded. This kind of beam-engineering bares an interesting application potential and will be investigated in detail in the following.

The confocal system point-spread function can be assumed to be a product of intensities:

Following [7] an image*p*

_{CL}obtained by confocal-line detection (CL) from a fluorophore distribution

*c*(

_{F}**r**) can be approximated bywhere the symbol ’*‘ designates a convolution. In Fourier space Eq. (8) is the convolution of the corresponding transfer functions:

*Ẽ*

_{ill}(

**k**) and

*Ẽ*

_{det}(

**k**) are the Fourier transforms of the illuminating field (Eq. (2)) and of the coherent detection PSF. AC{..} designates the auto-correlation.

Figure 6 displays the confocal transfer function *H*_{sys}(**k**,β) for various section angles. The first row illustrates projections of the angular spectrum *Ẽ*_{ill}(*k _{x}*,

*k*,β) along the beam propagation axis

_{y}*z*. The second row shows

*H*

_{ill}(

*k*β) = AC[

_{x},k_{y},*Ẽ*

_{ill}(

*k*β)], whereas

_{x},k_{y},*H*

_{sys}(

*k*β) is shown in the third row. From the shape of

_{x},k_{y},*H*

_{sys}(

*k*β), the beam’s sectioning capability can be estimated. It is roughly expressed by the ratio of high spatial frequencies (HSF) to medium/low high spatial frequencies (LSF) of the MTF. The LSF components lie close to the center of the axis which appears in yellow and white for all beams indicating the largest amplitudes. Medium amplitudes are shown in red and extend to the highest frequencies (i.e. largest distances to the center) for the conventional Bessel beam. The line-profiles shown to the right of Fig. 6 confirm that the Bessel beam MTF contains the highest spatial frequencies. However, the support for medium frequencies (see extent of yellow areas in

_{x},k_{y},*H*

_{sys}(

*k*)) is smaller in comparison to the sectioned Bessel beam for β = 0.6…0 8π. This finding indicates the different sectioning capability for sectioned Bessel beams as analyzed in detail below.

_{x},k_{y}The system’s point-spread function *h*_{sys}(**r**) (see Eq. (7)), can be measured in 3D when *c _{F}*(

**r**) corresponds to a small fluorescent sphere. The integral over

*h*

_{sys}(

**r**) is a measure for the total fluorescence signal. To quantify the optical sectioning performance we compute the total fluorescence as a function of the detection axis, such that

*H*

_{sys}(

*k*,

_{x}*k*) is the modulus transfer function of

_{z}*h*

_{sys}(

*x*,

*z*). The normalized integral of

*F*(

*y*),

*dy*

_{OS}of the optical section is defined such thatand gives the range along the detection axis out of which the majority of the fluorescence is collected. Alternatively, it is useful to investigate the signal-to background ratio SBG. Here the signal $S={\displaystyle {\int}_{-\delta}^{\delta}F(y)\text{d}y}$ is defined as the fluorescence generated within the depth of field of the detection lens 2δ ≈ 2λ / (n-n⋅cosα) and the background is the remaining fluorescence. Accordingly, the signal-to-background is

*G*(-δ)) /

*G*(-δ) for

*F*(

*y*) =

*F*(-

*y*).

## 6. Influence of the section angle β on optical sectioning

For optimum image quality we seek the angle β that provides the best optical sectioning. Therefore we evaluate the section thickness *dy*_{OS} for a series of angles β using numerically computed distributions of *h*_{sys}(*x*,*y*,*z*). The simulation was performed using a discretization of 23nm in a cube with a side length of 4096 pixels (≈94µm). The wavelength λ_{0} = 488nm corresponds to 8 pixels in a medium with *n* = 1.33. Examples for the cross-sections *h*_{ill}(*x*,*y*) of a sectioned Bessel beam for β = 20° and β = 110° and a conventional Bessel beam are shown in Figs. 7(a)-7(c). The ring parameter is ε = 0.9 so that the depth of field is *dz*_{1/e} ≈80µm. The detection point-spread function is shown in Fig. 7(d).

First, we analyze optical sectioning *dy*_{OS} for scanned illumination beams and wide-field detection in a light-sheet microscope. In this case, *dy*_{OS} depends only on the thickness of the light-sheet. The intensity of the light-sheet is constant along the scan axis (*h _{LS}* (

*y,z*) ∝ ∫

*h*

_{ill}(

*x*,

*y*,

*z*)d

*x*) and within the depth of field of the beam (

*h*

_{LS}(

*x*,

*y,z*) ≈

*h*

_{LS}(

*y*)), so that one can write

*F*(

*y*) = ∫∫

*h*

_{LS}(

*y*) ·

*h*

_{det}(

*x*,

*y*,

*z*) d

*x*d

*z*=

*h*

_{LS}(

*y*) · ∫∫

*h*

_{det}(

*x*,

*y*,

*z*) d

*x*d

*z.*As the detection lens collects equal amounts of light from each plane, i.e. ∫∫

*h*

_{det}(

*x*,

*y*,

*z*) d

*x*d

*z*=const, the proportionality

*F*(

*y*) ∝

*h*

_{LS}(

*y*) holds. The section thickness

*dy*

_{OS}(β) is shown in Fig. 7(j). With NA = 0.4 and ε = 0.9, a minimum thickness of

*dy*

_{OS}≈2.5µm is achieved for β = 30° and increases significantly for larger values. At the same NA, thinner light-sheets can only be generated by shorter beams having a smaller ε. For e.g. ε = 0.8 and a Bessel beam with

*dz*

_{1/e}≈40µm,

*dy*

_{1/}

*≈1.7µm is possible for β = 50°.*

_{e}However, there is a fundamental difference in the case of confocal-line detection. As indicated by Fig. 7(d), the detection objective lens collects light from a region that can be roughly approximated as a double cone with opening angle α_{det} = sin(NA_{det}/*n*). As described by Eqs. (7) and (10), the optical sectioning depends on the overlap between the illumination beam and the point-spread-function of the detection objective lens. This overlap can be steered for sectioned Bessel beams by adapting the value β relative to α_{det}. The system-point-spread functions *h*_{sys} corresponding to the beams shown in Figs. 7(a)-7(c) are displayed below in Figs. 7(e)-7(g) for a detection PSF with NA = 0.8 (Fig. 7(d)). It can be seen, that the small value of β = 20° results in a grating-like structure that is only poorly confined along *y*. Accordingly, *F*(*y*), shown in Fig. 7(h) is very broad. For the conventional Bessel beam (Fig. 7(e)), even though the main signal is confined well, the ring structure leads to strong sidelobes of *F*(*y*). The best result, i.e. a small extent of *h*_{sys} along the *y*-axis (Fig. 7(h)) and weak side-lobes is obtained for the sectioned Bessel beam with β = 100° (Fig. 7(f)). *G*(*y*) is plotted in Fig. 7(i) and exhibits the steepest and narrowest step for the sectioned Bessel beam. Figure 7(j) shows the optical sectioning *dy*_{OS} for various angles β = 0…180°. The thinnest sections are obtained for angles β in the range of 80° to 140°. Therefore, the optical sectioning for equal depth of field (ε = 0.9) is improved by a factor of ≈4 from *dy*_{OS} = 2.4µm to *dy*_{OS} = 0.6µm relative to conventional Bessel beams with confocal-line detection or sectioned Bessel beams with wide-field detection.

## 7. Optical sectioning dependency on the depth of field

The special angular spectrum of the sectioned Bessel beam cannot only be exploited to improve the optical sectioning, but also to make it independent on the depth of field, i.e. the size of the field of view along the illumination axis. This ability of the sectioned Bessel beam stands in sharp contrast to other illumination beams like Gaussian and Bessel beams, where the symmetrical cross-section increase for larger depths of field.

In Fig. 8 the optical sectioning for confocal-line detection is plotted against the illumination beam's depth of field for Gaussian, Bessel and sectioned Bessel beams. For the Gaussian beam a nonlinear dependency can be observed that is in agreement with Eq. (3c) predicting a square-root dependence between the Gaussian beam's depth of field Δ*y* and its waist diameter Δ*z* (Fig. 2(c)). Optical sectioning provided by a Bessel beam increases linearly, as can be seen by the good agreement of the values with a fitted line. The steps in the *dy*_{OS}(*dz*) originate from the Bessel beam’s rings. In contrast, optical sectioning for the sectioned Bessel beams is completely independent on the depth of field. The sectioned Bessel beam’s cross-section mainly increases on the sides (along the *x*-axis) where no light is collected by the detection lens, i.e. the only overlap between the illumination PSF *h*_{ill} and the detection PSF *h*_{det} is in the thin beam center around the beam’s axis.

## 8. Images of tumor multicellular spheroids

In addition to the analytical concepts and numerical simulations shown above, experimental results obtained from the imaging of cancer cell clusters are presented in this section. To test the performance of sectioned Bessel beams we acquired 3D stacks of Tumor Multicellular Spheroids. These Mouse colon carcinoma CT26 spheroids were prepared using a microfluidics-assisted extrusion device (Alessandri et al, to be published) and cultured during 4 days. They were fixed in 4% paraformaldehyde in phosphate buffered saline (PBS) during 1 h at room temperature and incubated with 0.5µg/ml Phalloidin conjugated with Alexa488 (Molecular Probes) in PBS with 1% v/v Triton-X100 (Sigma) at 4°C overnight. The spheroids of about 250µm diameter were mounted in 1% low melting point agarose (Invitrogen) in PBS.

We compare the performance of sectioned Bessel beam illumination combined with confocal-line detection with 3 other imaging modes: scanned Gaussian beam illumination as well as confocal-line detection with Gaussian beams and conventional Bessel beams. For each mode, a stack of 13 images with a spacing of 10µm was recorded. Figures 9(a)-9(d) show images *p*(*x*,*y*_{i},*z*) for *y _{i}* = 100 μm, located approximately in the center of the spheroid. The images reveal that confocal-line detection is able to improve visually perceived contrast well above the level of scanned Gaussian beams. However, in the case of Gaussian beam illumination, the signal for large penetration depth (right side) becomes very weak and the structure of the cell walls is hardly visible anymore. We quantitatively investigated the penetration depth by integrating the images shown in Figs. 7(b)-7(d), along the

*x*-axis to obtain

*p*

_{avg}(Fig. 7(e)). This measurement quantifies the beam’s on-axis power similar to the

*Q*-Values computed from numerical data in Section 4. We could confirm the simulation results and found that sectioned Bessel beams perform better than Gaussian beams and equally well as conventional Bessel beams regarding penetration into a scattering medium.

To quantify the image contrast, we computed the ratio of high-frequency and low-frequency image information. Similar as in [17], each image *p _{y}*(

*x*,

*z*) was Fourier transformed to $\tilde{p}({k}_{x},{k}_{z})=FT\left[p(x,z)\right]$ and the resulting spatial frequency domain image was separated into high (HSF, where

*k*= (

_{r}*k*

_{x}^{2}+

*k*

_{z}^{2})

^{1/2}≥

*k*) and low (LSF, where

_{F}*k*<

_{r}*k*) contributions separated by the corner frequency

_{F}*k*

_{F}= 1.1/µm. The ratio of the integrated high- and low-frequency contributions

*C*as previously in [7]. This expression is especially useful as it effectively describes the usable in-focus intensity signals (HSF) relative to the out-of-focus image intensity (LSF). Figure 9(f) shows the contrast

*C*(

*y*) for all image planes

_{i}*p*(

*x*,

*y*

_{i},

*z*). The measurement confirms the visual perception: confocal-line detection increases image contrast for all illumination modes. The contrast

*C*(

*y*) depends also on the plane

_{i}*y*of the image inside the spheroid which is due to scattering of the fluorescence on the detection side. Figure 9(f) also reveals that sectioned Bessel beams offer better contrast than conventional Bessel beams. The values

_{i}*C*(

*y*) are similar to Gaussian beam illumination. However by comparing Fig. 9(b) and 9(d) one can directly see that the sectioned Bessel beam yields a more favorable combination of overall image contrast and image detail at large penetration depths.

## 10. Discussion

The extents *dx*, *dy*, *dz* of a sectioned Bessel beam (SBB) in free space were described by evaluating the extents *dk _{x}*,

*dk*,

_{y}*dk*of its finite width ring spectrum in

_{z}*k*-space, which is mainly defined by the sectioning angle β and the ring width parameter ε. These criteria give a good estimate for the dimensions of the illumination beams when scattering is negligible.

To be able to compare the performance of different beams in scattering media we calculated their directional propagation stability. By computing the beam power within a small area around the beam propagation axis *Q* we were able to directly quantify many important beam properties that are of great interest to microscopy: the penetration depth into inhomogeneous media and the influence of the position of the beam in the scattering medium on the beam shape. Beams that show superior directional propagation stability are beneficial to light-sheet microscopy in two ways: an increased penetration depth and reduced artifacts. In contrast to the self-similarity of the transverse beam profile [18] our measure allows a direct comparison to the experimental data acquired in a light-sheet microscope using the confocal-line detection mode.

Directional propagation stability was observed in simulation data of coherent beam propagation in scattering media consisting of spheres with two different sizes and densities (Figs. 3-5). Thereby, the random distributions of the spheres extended over the whole cross-section of the beams. The Bessel beam with the higher NA and therefore larger cross-section is significantly more robust to scattering. However, the similar performance of the conventional and the sectioned Bessel beam indicates that the self-reconstruction ability depends on the radial component of the wave-vectors. For β = 88°, the sectioned Bessel beam exhibits a cross section that is only half as large as that of the conventional Bessel beam. While the conventional Bessel beam is affected by more scatterers, the relative weight of each is smaller than in the case of the sectioned Bessel beam. Therefore, in a medium where the scatterers are homogeneously distributed, the average perturbation of the phase of the Bessel beam and the sectioned Bessel beam over their cross-sections are similar.

In the images of tumor multicellular spheroids, sectioned Bessel beams offer similar signal-to-background as Gaussian beams when combined with confocal-line detection. However, especially in the back of the sample, sectioned Bessel beams offer significantly higher signal strength. We measured an increase in contrast over conventional Bessel beams of approximately 15%. Based on the data obtained by simulations (Figs. 7 and 8), where the sectioned Bessel beam was found to provide 4 times better optical sectioning, one would expect an even larger increase in signal-to-background. There are two explanations for the moderate increase in measured signal-to-background. First, as can be seen in Fig. 6, the conventional Bessel beam provides the highest frequencies along the detection optical axis due to its highly confined central lobe. Therefore, even though the background is higher, the conventional Bessel beam is able to illuminate the sample in close proximity to the focal plane of the detection lens with a high intensity and provide high spatial-frequency image details. Several details in Fig. 9 show that Bessel beams are able to reveal very fine structures within the samples that are blurred by Gaussian beam illumination even though these beams provide better overall signal-to-background. Second, the image of the structure itself is not entirely composed of high-spatial frequency components. It is very likely that both the Gaussian beam (for low penetration depth) and the sectioned Bessel beam illuminate only a thin slice within the sample and yield an image that is composed of a comparable ratio of high to low spatial frequencies that is limited not by the illumination but by the fluorophore distribution and scattering on the detection side.

In dependence of the section angle β, we investigated the amount of the potential photo-damage produced by SBBs, by analyzing that fraction of the beam that illuminates fluorophores only along the confocal line as previously in [4, 7]. The ratio η = *P*/*E* of the beam energy *P* within a circular region with radius *R* around the propagation axis *z* (*P*(*z*;*R*) according to Eq. (4)) and the total beam energy *E = P*(*z,∞*), is a useful measure for the fraction of the beam’s energy that excites fluorophores within the confocal volume. It is expected that this illumination efficiency η decreases for β<180°, especially for small values as for β = 20° (shown in Fig. 7(c)). However, within the range β>100° (Fig. 7(b)) that improves optical sectioning, the decrease in efficiency η in comparison to conventional Bessel beams (Fig. 7(a)) is small, especially for higher NAs and ring widths ε. For example, for NA = 0.4, ε = 0.8, the efficiency decreases by ≈30% for β = 100°, but for NA = 0.4 and ε = 0.9, the decrease of η for values of β>100° is only <10% in comparison to the conventional Bessel beam. These values were obtained from simulated data for *R* = 1µm. In conclusion, to maximize the efficiency η, the largest value of β that still enables good optical sectioning should be used. For example, as can be seen in Fig. 9, β = 140° already provides much better optical sectioning than the conventional Bessel beams for NA_{det} = 0.8. We expect that even higher values of β are suited for lower detection NAs that are often used for light-sheet microscopy of samples with a size of several hundred microns.

According to our simulations, the light efficiency η = *P*/*E* of a suitable SBB (large β) is slightly lower than that of a Bessel beam. However, the optical sectioning of a SBB is much better than that of a conventional BB. For example, for NA = 0.4 and ε = 0.9, one needs to increase the illumination intensity by 10% to gain a factor of 2 in sectioning (SBG) by using a SBB with β = 100°. This SGB value was computed according to Eq. (13) with *G*(*y*) shown in Fig. 7(i) for a depth of field of the detection lens of 4µm.

SBBs in wide-field detection with scanned illumination beams [3, 5] offer much better optical sectioning than conventional Bessel beams according to our computer simulations. Because no fluorescence is rejected by the slit detection, the signal *S* is created by all fluorophores within a slice around the focal plane of the detection lens (with thickness corresponding to its depth of field). For this case, using numerical data, we computed an efficiency η, that is approx. 3-5 times higher than for confocal-line detection [7]. However, the optical sectioning is generally worse for wide-field detection (Fig. 7(j)) resulting in lower signal-to-background. Nevertheless SBBs with angles β = 30°…50°, where optical sectioning is best to, could be used illuminate large samples that are sensitive to light exposure. This allows to exploit the deeper penetration into strongly scattering media in comparison to Gaussian beams (Fig. 4).

## 11. Summary and conclusion

In summary, we have introduced a new variation of a Bessel beam, the sectioned Bessel beam. By simulating and measuring its performance in strongly scattering media we found that the directional propagation stability is equal to that of conventional Bessel beams.

To achieve the best optical sectioning, sectioned Bessel beams should be used in a confocal-line detection light-sheet microscope. According to our computer simulations, optical sectioning is 3 times better than for Gaussian beams already at a depth of field of 100µm and even larger for more extended samples that are typically imaged in light-sheet microscopy. Since optical sectioning of sectioned Bessel beams with confocal-line detection is independent of the field of view along the illumination axis, these beams enable submicron optical sections even in very large. These advantages make sectioned Bessel beams an ideal candidate for illuminating large samples in light-sheet microscopy.

## Acknowledgments

We thank Cristian Gohn-Kreuz for helpful discussions. This work was financed by the Baden-Württemberg Stiftung.

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