A systematic and quantitative comparison of lattice and Gaussian light-sheets

The axial resolving power of a light-sheet microscope is determined by the thickness of the illumination beam and the numerical aperture of its detection optics. Bessel-based optical lattices have generated significant interest owing to their potentially narrow beam waist and propagation-invariant characteristics. Yet, despite their significant use in Lattice Light-Sheet Microscopy, and recent incorporation into commercialized systems, there are very few quantitative reports on their physical properties and how they compare to standard Gaussian illumination beams. Here, we systematically measure the beam properties in transmission of dithered square lattices, which is the most commonly used variant of Lattice Light-Sheet Microscopy, and Gaussian-based light-sheets. After a systematic analysis, we find that square lattices are very similar to Gaussian-based light-sheets in terms of thickness, confocal parameter and propagation length.


Introduction
Light-sheet fluorescence microscopy (LSFM) is a rapidly growing volumetric imaging technique that offers life scientists a unique combination of speed, sensitivity, optical sectioning, and low phototoxicity and photobleaching [1]. In LSFM, a biological specimen is illuminated with a focused sheet of light, and fluorescence originating from this sheet of light is captured in a widefield format using modern scientific cameras. Thus, unlike traditional epi-fluorescence imaging methods that nonproductively illuminate regions above and below the detection objective depth-of-focus, LSFM restricts the potentially damaging illumination to only the in-focus region of the specimen. This not only decreases phototoxicity and photobleaching, but delivers higher contrast imaging owing to its inherent optical sectioning capability [2].
Several factors critically influence the performance of LSFM [3]. Like conventional epifluorescence widefield microscopes, the lateral resolution of LSFM scales linearly with the fluorescence emission wavelength ( !" ) and inversely with the numerical aperture ( #!$ ) of the detection objective, !" ⁄2 #!$ . However, the axial resolution of LSFM depends upon the wavelengths of the illumination sheet ( !% ) and fluorescence emission ( !" ), and the numerical aperture of the excitation light-sheet ( !% ) and detection objective ( #!$ ). In the Fourier domain, this can be estimated from the highest axial frequencies that contribute to image formation, where #!$ is the solid angle over which the detection objective collects light and n is the refractive index of the medium. As a result, the axial resolution can be described as: ) &'( !" ) !" + *(,-./0 (2 #!$ ) ) !% + -, [3]. Of note, this is an optimistic estimate that assumes no attenuation of higher axial frequencies (e.g., optical transfer function roll-off), but it nevertheless serves as an upper bound for the axial resolution of LSFM. Consequently, for thick light-sheets, the numerical aperture of the detection objective largely dictates the axial

Experimental setup for Light-Sheet generation
A detailed schematic of the experimental setup is shown in Appendix A. The system allows the generation of optical lattices and Gaussian light-sheets and possesses a detection path with which these light-sheets can be imaged in transmission. In general, all lenses were placed in a 4f arrangement, which was confirmed with a shear plate interferometer, and the lateral and rotational alignment of each element was confirmed by measuring residual back-reflections from each optical surface. For all experiments described here, a 488-nm continuous wave laser (Sapphire 488-300 LP, Coherent) was used as the light source, which was spatially filtered by focusing it through a 50-μm pinhole (P50D, Thorlabs) with a 50-mm focal length achromatic lens (AC254-050-A, Thorlabs) prior to being diverted down one of two optical paths with a flip mirror.
The first path is a modified version of a LLSM [9], which has been described in detail elsewhere [15]. Here, the laser is recollimated with a 75-mm achromatic doublet (AC254-075-A, Thorlabs) prior to asymmetric expansion into a line profile with a pair of achromatic cylindrical lenses (68-160, Edmund Optics, and ACY254-200-A, Thorlabs). This line-shaped beam uniformly illuminates a narrow strip on a spatial light modulator (SLM, SXGA-3DM, Forth Dimension Displays) where binary phase holograms are displayed to generate the desired optical lattice. A polarized beam splitter (10FC16PB.3, Newport) and a half-wave plate (AHWP10M-600, Thorlabs) are placed in front of the SLM, and together with the SLM, form a reconfigurable lattice light-sheet generator. The polarization of the input laser light was adjusted to maximize the light-throughput through this unit. The diffracted light from the SLM is focused by an achromatic lens (AC508-400-A, Thorlabs) onto a custom-designed binary transmission mask (Photo Sciences, Inc.), which consists of multiple annuli of various sizes that serve to block the unwanted zeroth and higher-order (e.g., N > 1) diffracted beams, and to specify the inner and outer numerical apertures of the lattice light-sheet. After passing through the mask, the desired diffraction orders are de-magnified through two achromatic lenses (AC254-150-A and AC254-100-A, Thorlabs) and projected onto a mirror galvanometer (6215H, Cambridge Technology) for rapid lateral dithering of the illumination light-sheet. Thereafter, the light reflects off of a folding mirror, and is relayed to the back pupil of the 40X NA 0.8 illumination objective (CFI Apo 40XW NIR, Nikon Instruments) with an achromatic lens (AC-254-100-A, Thorlabs), a second flip mirror, and a tube lens (ITL-200, Thorlabs).
The second optical path is recollimated with an 80-mm achromatic doublet (AC254-080-A, Thorlabs), and is used to create conventional Gaussian-based illumination light-sheets. Here, the laser light is asymmetrically magnified into a line profile by a pair of achromatic cylindrical lenses (ACY254-050-A and ACY254-200-A, Thorlabs), and focused with a cylindrical lens (ACY254-050-A, Thorlabs) onto a mirror galvanometer (6215H, Cambridge Technology) that is conjugate to the focal plane of the illumination objective for rapid light-sheet pivoting [16]. Thereafter, the light is reflected with a folding mirror, and relayed to the back focal plane of the illumination objective with an achromatic lens (AC254-075-A, Thorlabs) and two tube lenses (ITL-200, Thorlabs). An adjustable slit, which is placed conjugate to the back focal plane of the illumination objective, is used to control the numerical aperture of the Gaussian illumination light-sheet. Besides the slit, we also used annular masks and placed the line beam at specific position to generate Gaussian light-sheets (see also Appendix B). We used these datasets to supplement our light-sheet measurements with the variable slit. We note that in either way (slit or annular mask), only the central flat-top portion of the Gaussian input beam remains: I.e. the intensity profile in the back focal plane is relatively uniform, instead of possessing a Gaussian profile. But we decided to use the term Gaussian light-sheet throughout the manuscript, as this is a very common terminology in the light-sheet field.
Both the LLSM and conventional Gaussian-based light-sheets were evaluated with a camera conjugate to the back pupil of the illumination objective. Here, a third flip mirror directs the laser light towards the camera (DCC1545M, Thorlabs), and two achromatic doublets (AC254-100-A and AC254-050-1, Thorlabs), and a neutral density filter (ND10A, or ND20A, Thorlabs), served to relay and attenuate the laser power, respectively. For LLSM, the correct placement of the diffraction orders in relation to the annular mask was monitored and the inner and outer numerical apertures of the illumination beam were measured. For conventional Gaussian-based LSFM, the width of the line profile in the back focal plane was measured to determine the numerical aperture of the illumination.

Light-Sheet Measurements
To measure the characteristics of the light-sheets in the front focal plane of the illumination objective, the light-sheets were imaged in a direct transmission mode using another 40X NA 0.8 objective (CFI Apo 40XW NIR, Nikon Instruments). Here, a 400-mm focal length achromatic lens (AC508-400-A) served as the tube lens, and the laser light was detected with a scientific CMOS camera (Orca Flash 4.0 V2, Hamamatsu Corp.) after attenuation with a neutral density filter (ND40A, Thorlabs). The combination of a 400-mm tube lens and the 40X detection objective offers a final magnification of 80X, which results in a pixel size of 81.25 nm. To acquire three-dimensional data sets of the light-sheets, the detection objective was scanned along the direction of laser propagation with a piezoelectric stage (P621.1CD, Physik Instrumente). The data acquired encompassed the main in-focus region of the light-sheet, as well as the more diffuse out-of-focus regions before and after it (See Figure 1).
Importantly, it should be noted that all optical imaging systems generate a slightly blurred image of the real object (here, the light-sheets), which also holds for our transmission imaging setup. However, because the light-sheets that we have measured have a lower numerical aperture (0.19 -0.64) than our detection objective (0.80) and hence were significantly thicker than its diffraction limit, we expect this effect to be negligible. Furthermore, both the Gaussian and LLSM light-sheets were measured identically, and thus they are subject to the same amount of optical blurring. Lastly, by measuring the light-sheets in transmission, we argue that this avoids potential challenges that could arise from measuring the intensity distribution of the light-sheet with sub-diffraction fluorescent nanospheres (e.g., bead aggregation, saturation, bleaching etc.).

Light-Sheet Analysis
All light-sheet measurements were performed in a fully automatic fashion using a publicly available software written in MATLAB (https://github.com/AdvancedImagingUTSW). Since the light-sheet stays constant over hundreds of microns in the transverse direction (labeled X in Figure 1, we subdivided the acquired data into 6 sub-volumes along the X-direction and analyzed each block individually. This allowed us to confirm that the light-sheet was uniform in the X-direction and allowed for averaging of the measured parameters.  3) The confocal parameter, for which we measure the distance between the two points along the Y-direction where the light-sheet thickness increases by a factor of √2 from its smallest value.
To measure the light-sheet thickness, we first evaluated each XZ plane along the Y-axis to find the focus of the light-sheet, i.e. the location of the highest beam intensity ( Figure 2A). Next, the image dataset was rotated (usually less than 1°) such that the light-sheet was perfectly aligned perpendicular to the Z-axis ( Figure 2B). Figure 2C shows a line profile of the intensity along the Z-axis along the center of the light-sheet. We measured the light-sheet thickness from the standard deviation (σ) of a Gaussian function fit ( = 262 ln (2) ). Importantly, a Gaussian function might be a poor approximation for light-sheets with strong sidelobes (e.g., Bessel and Hexagonal lattices) [14]. However, we found that at the beam waist, the profile was well approximated by a Gaussian function for both the Gaussian light-sheets as well as the square lattice light-sheets ( Figure 2C).
To measure the propagation length, we first average the plane-by-plane image (YZ slice) along the X-axis for each block, and then we rotate the image slightly (usually also less than 1°) to make sure the propagation of the light-sheet is parallel to the Y-axis. Figure 2D shows a line profile of the intensity along the Y-axis along the center of the light-sheet. We measured the beam length from the standard deviation (σ) from the fit of a Gaussian function in the same way as we do for the light-sheet thickness. To measure the confocal parameter of the light-sheet, we measured the width of the light sheet in the Z-direction at multiple positions along the Y-axis, i.e. the intensity profiles of the white dashed lines in Figure 2A. Although a Gaussian curve fit provided satisfactory results near the beam waist, it failed in regions away from the light-sheet focus as the intensity distribution becomes increasingly non-Gaussian. Thus, we numerically estimated the FWHM of the light-sheet thickness by interpolating where the intensity drops to 50% of its peak value ( Figure 2E). As can be seen in Figure 2F, the light-sheet thickness remained relatively constant over a length ~20 microns but grows rapidly outside of this range. From this curve, we then numerically identified the confocal parameter, which is defined as two times the distance over which it takes the light-sheet thickness to increase by a factor of √2 (red stars in Figure 2F). We wanted to emphasize that for a purely Gaussian beam, the confocal parameter would be twice the Rayleigh length, which can be computed analytically from the beam waist thickness. However, we have observed that our light-sheets do not behave exactly like Gaussian lightsheets, i.e. their beam waist stays constant in the main lobe of the light-sheet, and then increases rather rapidly outside, yielding poor results if fitted with a quadratic function. As such, we decided to report the confocal parameter as defined by the √2 thickness increase.  [14]. For all simulations, a refractive index of = 1.33 and an excitation wavelength of !% = 0.488 was assumed. We decided to use their definition of main-lobe thickness ( "6 ) and Length, which are output parameters of the simulation. As previously noted, the square lattice contains small sidelobes, but they are small enough that "6 can effectively capture the thickness of it. "6 is defined as the beam size when the intensity drops by a factor of 1⁄ . Thus, the factor ≅ 0.8326 × "6 is applied to the simulation results. We also include the theoretical values for Gaussian light-sheets using an analytical formula for Gaussian beams [17]. The beam waist, 5 , used in Gaussian Optics is the radius of the beam when the intensity drops to 1 & ⁄ , and relates to the FWHM in the following way: 7 ≅ 0.85 × . By applying these relationships, the thickness (given as FWHM) and the confocal parameter (2 4 ) are obtained.

Systematic Evaluation of Light-Sheet Properties
To evaluate the optical properties of the square lattice light-sheet in LLSM, we systematically varied the inner (na) and outer (NA) numerical apertures of the illumination annulus and measured the thickness and propagation length of the light-sheet in transmission. In each case, a matching hologram was applied to the SLM, and the correct diffraction pattern was confirmed with the camera that is conjugate to the back focal plane of the illumination objective. We further evaluated parameters linked to the hologram generation, which among other things influenced the relative positioning of the diffraction orders relative to the annular mask. Lattice light-sheets were rapidly dithered in the X-direction with a mirror galvanometer to eliminate the lateral interference patterns. To facilitate a side-by-side comparison, we report measurements on nearly all of the square lattice light-sheets from the original Lattice light sheet microscopy manuscript [9]. We then acquired an exhaustive series of Gaussian light-sheets where the excitation NA was varied in small steps. From this data set, Gaussian light-sheets were selected that closely matched their lattice light-sheet equivalents. A list of the NAs for the measured Gaussian and square lattice light-sheets is provided in the Appendix C.
Qualitative differences are readily visualized in the cross-sectional views for the dithered square lattice and Gaussian light-sheets ( Figure 3). For example, the dithered square lattice, while having a similar shaped central lobe compared to the Gaussian sheet, exhibits some structural differences in the transition zones before and after the light-sheet focus (white arrows). In addition, the dithered square lattice features two distinct sidelobes (white arrows in XZ-view), whereas the Gaussian sheet is almost free of sidelobes. Further, in Figure 3A-B, one can see two faint beams coming from much steeper angles in the cross-sectional view for the square lattice, which are not present in the Gaussian sheet. These beams originate from the central diffraction order of the hologram used for lattice generation. Interestingly, we experimentally blocked these orders in Appendix D and found that they negligibly influence the properties of the light-sheet. In an effort to facilitate a more detailed assessment of light-sheet properties, Figure 4 provides detailed profiles for three square lattices and three Gaussian light-sheets with a matching confocal parameter. Strikingly, Figure 4A shows that both light sheets maintain a nearly constant thickness throughout their optical focus (e.g., within the range dictated by their confocal parameters). This is a property that was expected for the lattice light-sheet, as it derives from Bessel beams, which in turn have a constant intensity profile over a finite propagation length. It is however a surprise for the Gaussian light-sheets, which were expected to show a quadratic increase in beam waist thickness over their length. Importantly, the cross-sectional profiles ( Figure 4B) show a close correspondence in their main lobe thickness, which influences the axial resolution of LSFM. The square lattice is accompanied with side lobes that grow in intensity with increasing confocal parameter, more dominantly so than for the Gaussian lightsheet. Thus, for a given light-sheet confocal parameter, Gaussian beams appear to provide similar axial resolution, albeit with reduced image blur and greater optical sectioning capability. A more quantitative analysis of the relationship between light-sheet thickness and propagation length is summarized in Figure 5. Here, each point represents the average of multiple measurements obtained independently from different image sub-volumes of the lightsheet image data, and detailed values of the average and standard deviation are provided in Appendix C. As can be seen, and in agreement with previous numerical simulations [14], the tradeoff between light-sheet propagation length and thickness is similar for both square lattices and Gaussian light-sheets. Of note, the data points for the square lattice show greater variability than those for the Gaussian light-sheets. This arises owing to the larger number of degrees of freedom for generating square lattices. For example, we found that a light-sheet generated from a low NA thick annulus is similar to one generated from a high NA thin annulus (Appendix E). Furthermore, the SLM hologram has parameters that can be tuned, most notably the cropping factor and the spacing, and varying these factors does slightly alter light-sheet properties (Appendix F). While we have used the parameters that have been reported in the lattice literature, one cannot exclude the possibility that through a judicious choice of parameters, A. Beam size along propagation B. Light-sheet profile at focus more advantageous square lattices could be generated. Nonetheless, in our hands, the best-case lattice light-sheets (i.e. possessing a minimal thickness for a given length) did not outperform the Gaussian light-sheets but followed an almost identical trend for thickness increase with increasing confocal parameter. Furthermore, the data suggest that the inherent complexity of LLSM likely results in greater performance fluctuations than a microscope equipped with a comparatively simple Gaussian light-sheet.

Comparison of Simulated and Experimentally Measured Light-Sheets
Recently, Remacha et al numerically evaluated the optical characteristics for a diverse array of light-sheets (including Gaussian, Bessel, dithered lattice, airy, and more) and concluded that Gaussian beams provided superior image contrast [14]. However, these data were not experimentally confirmed, which is of particular concern for lattice light-sheets since the simulations assumed that each diffraction order in the back focal plane of the illumination objective had constant phase and electric field strength. As such, it has remained an open question if such simulations correctly predicted the properties of experimental lattices. Here we address these concerns and compare our experimental data to theoretical Gaussian beams, and simulated lattice and Gaussian light-sheets. In an effort to maintain consistency, the simulations were performed as described previously [14]. As can be seen in Figure 6, our experimental Gaussian light-sheets are thinner than the theoretical values for Gaussian beams. As mentioned earlier, the intensity distribution in the back focal plane of our illumination objective is more uniform than a traditional Gaussian beam and thus resembles a "flat-top" beam, which could potentially contribute to this discrepancy. Nevertheless, previous simulations by Remacha et al demonstrate that Gaussian and "flat-top" beams behave similarly [14], which we performed as well. Indeed, our experimental Gaussian light-sheets are in close correspondence with both the simulated Gaussian and "flat-top" beams, and overall, the difference appears small (Appendix G). Furthermore, both the simulated and experimentally measured "best-case" square lattice light-sheets are in near-perfect agreement, suggesting that the numerical simulations performed by Remacha et al were accurate. And lastly, our Gaussian beams are largely indistinguishable from the lattice light-sheets in terms of light-sheet thickness. Importantly, this analysis only takes into account the thickness of the main lobe, while ignoring the sidelobes that are present in the lattice light-sheets. Fig. 6. Comparison of experimentally measured Gaussian and square lattice light-sheets with numerically simulated Gaussian, square lattice, and theoretical Gaussian light-sheets derived from an analytical formula.

Summary
LLSM has generated significant interest in the biological imaging community, with several high-profile publications referring to its illumination optics as "ultrathin" [9][10][11]18] and its resolution as "unparalleled" [19]. Yet, as we have shown previously [15], the time-averaged equivalent of these illumination beams can be generated incoherently, and therefore their performance is not a unique consequence of the coherent superposition of multiple Bessel beams. Furthermore, as we show here, the most popular lattice light-sheet, the dithered square lattice, does not appear to provide a significant performance improvement over Gaussian lightsheets, as measured from their beam waist thickness and confocal parameter. We also investigated the Optical Transfer Function for both Gaussian and square lattice light-sheets, and found that both Optical Transfer Functions extend to a similar level (Appendix H). Thus, for LSFM, these beams would result in practically indistinguishable levels of optical sectioning, field of view, and axial resolution. According to both numerical simulations [8,14] and the experimental measurements, the square lattice does not behave like a propagation invariant beam, but rather a divergent one. Indeed, its tendency to increase in thickness as the confocal parameter grows is in close agreement to a simple Gaussian light-sheet. This in itself is a surprise, as square optical lattices are produced by a coherent superposition of Bessel beams, which are de facto propagation invariant. Therefore, we conclude that for certain periodicities of the lattice (i.e. corresponding to the square lattice), the propagation invariant nature of the parent beam can be lost, and a clear explanation for this phenomenon is still under investigation.
For higher resolution imaging, the coherent modulation present in static (e.g., non-dithered) lattice light-sheets can be used for structured illumination microscopy [9]. However, owing to the LSFM geometry, this mode improves the resolution only in the axial dimension and one of the two lateral dimensions (e.g., the X-direction). As such, we believe that the main benefit of optical lattices for structured illumination microscopy is that they provide greater frequency support in the axial dimension which cannot otherwise be achieved yet with one dimensional beam engineering. Nonetheless, owing to the need to acquire 5 images for one structured illumination reconstruction, and the fact that the lateral resolution gain is modest, the use of this LLSM mode has so far been rare. Alternatively, hexagonal lattices provide a thinner main lobe than square lattices at the cost of much stronger sidelobes that grow drastically with the light-sheet propagation length. Indeed, sidelobes that approach 50% strength of the main lobe are notoriously hard to remove computationally [20], which is likely the main reason why hexagonal lattice light-sheets have seen little use [21]. And lastly, it appears to us that both for the square and hexagonal lattices, the reported resolutions could only have been achieved via strong deconvolution of the data.
In conclusion, as the vast majority of LLSM is performed with the dithered square lattice and not the structured illumination or hexagonal lattice light-sheet modes, our results stipulate that this type of imaging could alternatively be performed with conventional Gaussian or flattop light-sheets. This would be a welcome simplification of such high-resolution light-sheet microscopes, as the correct alignment of a LLSM setup requires optical expertise. Furthermore, by dispensing with the spatial light-modulator, much less input laser power is needed, and simultaneous multicolor excitation is also easily achieved. Importantly, we interpret our results as a poignant reminder that there is no free lunch in linear optics; with increasing confocal parameter, it appears that one either has to accept a thicker light-sheet, stronger sidelobes, or both. Lattice light-sheets have been heralded as a breakthrough that overcomes this trade-off. Nonetheless, our results indicate that they underlie the same physics and limitations as a simple Gaussian light-sheet.

R.F. receives funding from the Cancer Prevention Research Institute of Texas (RR160057) and
the National Institutes of Health (R33CA235254 and R35GM133522). K.M.D. receives partial salary support from 5P30CA142543.

Acknowledgments
The authors would like to thank Florian Fahrbach for help with the simulation code and critical reading of the manuscript. Furthermore, we are grateful to Dr. Kim Reed for her continued support, as well as all the employees at the University of Texas Southwestern Medical Center who make our research possible.

Article thumbnail upload Appendix A. Experimental System
Fig. S1. Schematic representation of the experimental setup, which consists of two illumination paths, one to produce optical lattices (blue arrow "Lattice LSFM") and one to produce conventional Gaussian light-sheets (green arrow "Conventional LSFM"  Line beam

Appendix C. Systematic Comparison of Gaussian and square lattice Light-Sheets
Table S1 summarizes all Gaussian and square lattice light-sheets that we have measured. In total 39 and 52 experiments have been performed for Gaussian and square lattice light-sheets, respectively. The data covers a wide range to illustrate the characteristics of both Gaussian and square lattice light-sheets. In the square lattice, NA and na represent the outer and inner numerical apertures of the annulus, respectively. Bold indicates that the light-sheets were used in the original Lattice Light-Sheet Microscopy manuscript [14]. For Gaussian beams, the effective NA used to generate the light-sheet was controlled in most cases with a micrometer slit that is conjugate to the back-pupil of the illumination objective. In some cases, we replace the micrometer slit with an annulus. In such case, the annulus served as the slit, and the effective NA is reported as described in Figure S2. Table S1. Summary of all experimental data shown in Figure 5. The table is sorted with thickness and then the confocal parameter. Eff. NA represents the effective NA used for Gaussian beams, which corresponds to the NA of the slit or the theoretical NA of the annulus. In the square lattice, sp and cp represents spacing and cropping factor, respectively. n represents the number of measurements. For Gaussian measurements, n equals to 24 for all measurements. In the square lattice measurements, n equals 6 unless it is specified otherwise.

Appendix D. Contribution of Zeroth Order Beams to Square Lattice Properties
We performed measurements on square lattices before and after blocking the zeroth, +1, and -1 diffraction orders. Only one data set is shown for each annulus, however similar results can be found with other measurements. Table S2 summarizes the measurements with multiple annuli. It can also be seen that when the 0th order is blocked or only +1 or -1 is allowed, the light-sheet still shows similar properties to the original dithered square lattice light-sheet

Appendix F. Choice of SLM Parameters for Lattice Generation
The SLM phase hologram is essential for generating lattice light-sheets. The code we are using to generate the SLM phase hologram basically follows the practical optical path. It first generates the pattern on the back focal plane (BFP) of the illumination objective by considering that the lattice pattern is the interference of a group of Bessel beams. The Fourier transform of the pattern on the BFP of the illumination objective is then the basis of the pattern put on the SLM. The SLM pattern is affected by the size of the annulus, the wavelength, and the magnification of the optical path. Nevertheless, to obtain a proper lattice, several parameters need to be adjusted carefully, and the cropping factor and the spacing are the two main parameters. The cropping factor is basically a threshold that crops some residues in the SLM phase hologram. In practice, it affects the sidelobes in the final lattice light-sheet. As shown in Figure  S5 indicated with the white arrows, with larger cropping factor, the sidelobes and the intensity of the short line-beams are decreasing, and the lattice becomes more Gaussian like. Please also note that in Figure S5, spacing = 0.94, large sidelobes appear. So even with very large cropping factor it does not resemble anymore to a Gaussian light-sheet. Nevertheless, we can still see the side-lobes are decreasing with larger cropping factor.
The spacing parameter basically determines the period of the lateral periodic pattern of the lattice. Effectively, it also determines the position of the long line-beams, or the distance between them, as indicated by the red dashed line in Figure S5. Generally, the spacing parameter is chosen when the long line-beams are in the middle of the outer and inner rings of the annulus. However, there is still some flexibility in their fine spacing. Basically, the larger the spacing parameter, the closer the two line-beams, and the light-sheet is also shorter because the long line-beams are getting longer, as explained in Figure S4(B and C). Please also note that for a spacing = 0.94, strong sidelobes that are characteristic for hexagonal lattice patterns start to appear. Table S3 shows the quantifications of the light-sheets shown in Figure S5, which also supports the aforementioned observation. We also put two Gaussian light-sheet that has similar thickness for comparison, and they show similar properties to the dithered square lattice lightsheet. We can see that with the same thickness (~1.5 µm), the Gaussian light-sheet has a length (~62-64 µm) that is very similar to the longest dithered square lattice (spacing = 0.92, cropping factor = 0.05) in Figure S5. The light-sheet length increases (69.8 µm) along with an increase of its thickness. We did not include the dithered square lattice light-sheet with spacing= 0.94 because it behaves more like a hexagonal lattice light-sheet and the quantification of it is very different. The XZ slice of the non-dithered lattice is shown in the rightmost column. Note that the scaling in the XZ slice is different from the YZ in order to show clearer the lattice pattern and the side lobes. In the BFP, white arrows point to the short line beams and how they are affected by the cropping factor (cp). The red dashed line indicates the shift of the long line beams depending on the spacing. In the YZ and XZ slices, white arrows point to the appearance and disappearance of the sidelobes. The NA and na of the annulus used in this example are 0.55 and 0.52, respectively.    [14].