1. Introduction

Light sheet fluorescence microscopy (LSFM) is a novel 3D imaging technique that can obtain a high temporal-spatial resolution with the advantages of rapid acquisition and 3D low-damage imaging [1–3]. It has been widely used in developmental and cellular biology, anatomy and neuroscience [4–7]. The traditional fluorescence microscope uses a same objective to perform illumination and detection simultaneously. In contrast, the light sheet fluorescence microscope uses separate orthogonal excitation and detection light paths. Its excitation light path produces a thin sheet-like optical field, known as a light sheet, that illuminates the sample from the side [8–10]. The excited fluorescent signal is collected and imaged by a detection light path perpendicular to the light sheet. The fluorescence is only excited in the illuminated plane, reducing the out-of-focus fluorescence background and thereby improving contrast. Also, as the light sheet illuminates only one sample plane at a time, the time for biological samples to be irradiated is significantly reduced, thus reducing photobleaching and phototoxicity [11].

The optical sectioning capability of light sheet fluorescence microscopy is related to the thickness of the light sheet [4]. The axial resolution can be significantly improved if the light sheet is thinner than the axial extent of the detection objective point spread function (PSF). Even if it is thicker than the axial extent of the detection objective PSF, the resolution can still be improved in the thick fluorescent sample. This is because the resolution is influenced by the contrast of the image in practice [11]. The traditional light sheet microscope uses a static 2D Gaussian light sheet which is formed by focusing with a cylindrical lens. However, an unavoidable problem is a tradeoff between the light sheet thickness and propagation length, which is due to the rapid divergence of the Gaussian light sheet [12,13]. Higher axial resolution is often gained at the expense of a smaller field of view (FOV). Another problem is that the static light sheet can induce serious artifacts. To overcome these problems, different methods have been proposed to relieve the trade-off between the axial resolution and FOV, such as multi-view imaging technique [14], tiling light sheet microscopy [15], axially swept light sheet microscopy [16] and others [17]. However, these methods lack of direct improvement of the performance of the light sheet illumination. More direct approaches such as Bessel beams [18,19], Airy beams [20], and lattice light sheet (LLS) [21] explore for the possible best theoretical solution of the light sheet illumination. The Bessel beam and the Airy beam , which are self-healing and diffraction-free, were dynamically scanned to form a virtual light sheet. But these virtual light sheets have strong side lobes in the axial direction of the detection objective, which excite out-of-focus background and significantly affect the axial resolution. In 2014, lattice light sheet microscopy (LLSM) has been first proposed by Eric Betzig’s group [21]. This technique generates structured light sheets by bounded 2D optical lattices, or coherent Bessel beam arrays. The introduction of the such a structured light sheet further balances the light sheet thickness and perpendicular length. Compared to the scanned light sheet using Bessel or Airy beam, LLS contains fewer side lobes energy, which leads to fewer phototoxicity and weaker background for vivo imaging.

LLS is a kind of structured light sheet which uses bounded 2D optical lattice patterns with both lateral (perpendicular to the propagation direction) and axial structures. LLS microscope can operate in two different modes: a high speed dithered mode and a super-resolution structured illumination microscopy (SR-SIM) mode. In the SR-SIM mode, optical lattice scanning is combined with the SR-SIM method [22]. Extended resolution beyond the diffraction limit in the directions perpendicular to the illumination pattern stripes can be achieved but with slow imaging acquisition and reconstruction speeds. In contrast, dithered mode provides faster imaging speed and is more suitable for in vivo imaging. In the dithered mode, a uniform light sheet is formed by continuously and rapidly dithering the lattice pattern. Although a LLS created in dithered mode has an improved tradeoff between axial resolution and FOV compared to Gaussian and Bessel light sheet, the out-of-focus background excited by the side lobes still affects the imaging quality. Various techniques, such as compressive sensing [23], deconvolution [24,25], and deep learning [26,27], are combined with fluorescence microscopy to improve image quality further. However, there are relatively few methods to improve the image quality directly by suppressing the side lobes of the dithered LLS. In 2021, Bin Cao et al. presented 3D interferometric lattice light sheet imaging (3D-iLLs) [28]. This technology provided effective background suppression and substantially improved volumetric resolution through the combination of 4pi interferometry and light sheet microscopic illumination. However, the combined use of three objectives in the system requires more sensitive optical calibration than LLS microscopy and further limits the space available for sample mounting. Also, this method requires higher stability for long-time localization-based super-resolution imaging. In 2022, Yuxuan Zhao et al. developed a double-ring (DR) Bessel light sheet system [29]. The energy ratio of the side lobes to main lobe of a DR line Bessel light sheet with a sheet thickness of 0.48 ${\mathrm{\mu} \rm {m}}$ was 45% lower than that of an LLS of the same thickness. Although the DR Bessel system effectively suppresses the light sheet side lobes and provides better axial resolution, the modulation between the optical fields by the DR leads to a certain sacrifice of the propagation length of light sheet.

In this paper, we present an effective method to suppress the side lobes of double-line light sheet (DLLS) [30], a simple kind of structured light sheet, by superposition of two light sheets with different side lobe distributions. The main lobes of the two light sheets interfere constructively, while the side lobes interfere destructively when the oscillation period of side lobes of one light sheet (DLLS 1) is different from another (DLLS 2). As a result, the main lobe is enhanced while the energy of side lobes is averaged and appears relatively small. Thus, the energy ratio of the main to side lobe can be greatly improved, i.e., the side lobes of the resulting light sheet (DLLS 3) can be effectively suppressed. The suppression effect of the side lobes is found to be determined by the length ratio of the line beams in the rear pupil plane which generate the two interfering light sheets. We estimate such a length ratio value using theoretical calculations. The simulation results indicate that, compared to the DLLS 1, the side lobes energy of the new DLLS 3 is reduced by more than 81.3%, the axial FWHM is slightly increased by 22%. Our proposed method effectively offers the possibility to suppress side lobes and can also achieve the purpose of suppressing the side lobes and improving the axial resolution for other types of the structured light sheet. Using the same strategy of superposition of two light sheets, the side lobe suppression of the four-line light sheet (FLLS) [28] and Line Bessel sheet (LBS) [31] are further explored separately. Compared to the case of the double-line light sheet, there is no significant reduction in the total energy ratio of the side lobes for the four-line light sheet and Line Bessel sheet. But the distributions of the side lobes are dramatically changed, and the levels of the first side lobe are efficiently reduced, which also results in enhanced imaging contrasts. Compared to existing side lobe suppression techniques, the proposed method is more straightforward in design and can be directly used in existing light sheet fluorescence microscope systems.

2. Theory and methods

The light sheet can be generated by a beam shaping system using a spatial light modulator (SLM) in the traditional LLS microscope [21] or via the field synthesis method [32] (not shown in Fig. 1). In both these two illumination methods, a specific illumination pattern has to be generated at the rear pupil plane of the illumination objective. Figure 1(a) shows the schematic of a simplified light path for generating light sheet illumination and orthogonal detection. For simplicity, we take a DLLS as an example. A structured light sheet can be created by projecting a pair of straight-line beam patterns at the rear pupil of the illumination objective, as shown in Fig. 1(a)-a1. Specific illumination line beam pattern is filtered by an annular mask with a desired inner and outer NA. These filtered line beams at the rear pupil of the illumination objective and the illumination optical field at the front focal plane correspond to the inter-conversion between the spatial frequency domain and spatial domains [33]. To meet the requirements of both high axial resolution and large FOV, the light sheet should extend as long as possible in the $x$ and $y$ directions to provide a large illumination region; meanwhile, it should be as confined as possible in the $z$-direction to reduce out-of-focus background and enhance axial resolution [33]. $k_x$, $k_y$ and $k_z$ are the wavevector components of the refracted light wave vector $k_0$ ($k_0 = 2\mit \pi /\lambda$) near the focal point of detection objective in the $x$, $y$, and $z$ directions, respectively. For a light sheet generated by double-line beams, these wavevector components have satisfied these criteria to a large extent with $k_z$ span range (corresponding to the line length) designed as large as possible and $k_x$ span range (corresponding to the line width) as narrow as possible.

 

Fig. 1. (a) Schematic diagram of a simplified light path for the structured light sheet. (b) A new light sheet with much weaker side lobes (b3) resulted from the superposition of two light sheets with different side lobe distributions (b1 and b2) along the thickness direction. The curves are the $z$-directional amplitude profiles of the dithered light sheets.

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The resulting light sheet achieves an optimized balance between axial resolution and FOV. However, the presence of side lobes still limits the axial resolution. Herein, we present an approach to suppressing side lobes by superimposing two light sheets with different side lobes distributions. The light sheet side lobe distribution is determined by the length of the line beams. Two light sheets with different side lobe distributions are designed by changing the line beam length, as shown in Fig.1(a)-a1 and a2. By projecting a composite illumination beam pattern which consists of a1-line beams and a2-line beams at the rear pupil of the illumination objective, as shown in Fig. 1(a)-a3, a new light sheet DLLS 3 can be resulted from the superposition of DLLS 1 (created by a1-line beams) and DLLS 2 (created by a2-line beams). The destructive interference occurs between the side lobes of DLLS 1 and 2, while constructive interference occurs between the main lobe of these two light sheets. Consequently, the resulting new DLLS 3 with strongly suppressed side lobes can be obtained. The line curves in Fig. 1(b)-b1 and b2 correspond to the cross-sectional profiles of the dithered light sheets along the $z$-direction, which are separately produced by the illumination beams in the rear focal plane of the illumination objective shown in Fig. 1(a)-a1 (a1-line beams) and Fig. 1(a)-a2 (a2-line beams). It can be seen that there are obvious side lobes in the cross-sectional profiles of the light sheets 1 and 2. An optical interference effect due to a superposition of the two light sheets in Fig.1(b)-b1 and b2 can occur and leads to a new light sheet, DLLS 3, as shown in Fig. 1(b)-b3. When the first positive peak of the DLLS 1 (A point) in b1 is overlapped with the first negative peak of the DLLS 2 (B point) in b2, the new DLLS 3 resulted from the interference between the two light sheets can own much weaker side lobes. We designed a3-line beams in the rear focal plane of the illumination objective to achieve this, as shown in Fig. 1(a)-a3. The a3-line beams are a superposition of the a1-line and a2-line beams, and the $z$-directional optical field profile of the new corresponding light sheet is shown in Fig. 1(b)-b3.

According to the principle outlined above, the suppression effect of the side lobes dramatically depends on the positions of the side lobes of the DLLS 1 and DLLS 2 in the $z$-direction. Considering that the optical field distribution of the structured light sheet is determined by the line beams in the rear pupil of the illumination objective, we Fourier transformed the line beams in the $z$-direction to obtain the $z$-directional cross-section field profile of the corresponding dithered light sheet in the focal plane. We define $L_1$ as the length of a1-line beams in the $z$-direction and $L_2$ as the length of a2-line beams. The distribution of these line beams in the $k_z$ direction can be regarded as a rectangular function. The Fourier transform of a rectangular function is a $sinc$ function. To achieve the side lobe suppression effect through light sheets superposition, the A point of the intensity cross-section profile of DLLS 1 in Fig. 1(b) is designed to be overlapped with the B point of DLLS 2. With a given $L_1$, we can calculate the position of the A(B) point by Fourier transform, and then the $L_2$ value can be deduced through an inverse Fourier transform. This offers a fast, convenient method for determining the setting of $L_2$ with very acceptable results, even if not optimal. To confirm such a method, we will obtain an optimal setting of $L_2$ for side lobe suppression by numerical simulations in the following results and discussion section.

Now, we analyze the Fourier transform of the line beams in the $z$-direction in the rear pupil plane of the illumination objective to obtain the $z$-directional intensity cross-section profile of the resulting light sheet. We use the following conventions: coordinates in the rear pupil plane $y = 0$ are denoted $z'$; and $z$ in the front focal plane $y = 2F$ ($F$ is the focal length of the illumination objective). The line beams distribution along $z$ in the rear pupil plane can be expressed in terms of:

With

The obtained $f_{2F}$ is the amplitude profile of the optical field distribution along $z$-direction, which equals zero when $z$ satisfies:

Then we can calculate the $z$ position of point A $z_A$ at the first positive peak of DLLS 1 shown in Fig. 1(b)-b1:

Similarly, the $z$ position of point B $z_B$ at the first negative peak of DLLS 2 shown in Fig. 1(b)-b2 is obtained as

According to our design principle of suppressing side lobes by superimposing two light sheets, point A and B should coincide, i.e.,

That is

Finally, the ratio of $L_2$ to $L_1$ for the side lobe suppression of DLLS should be

Another aspect we need to consider is that the energy matching of the two interfering light sheets. To obtain strong destructive interference of side lobes, the amplitude at A and B points should be identical or approximate. For simplicity, here the incident energy of a1 line beams is set equal to that of a2 line beams (energy matching). The incident energy of the line beams can be controlled through beam intensity modulation or the change in the beam illumination area. In our paper, the incident line beams are assumed to have identical intensity, and the width ratio of a2 to a1 line beams is changed to satisfy the energy matching condition. Hence, in our designs, $W_2/W_1 = L_1/L_2$ ($W_1$ and $W_2$ are the widths of a1-line beams and a2-line beams, respectively). Also, it should be noted that, in the $z$-directional overlapping region of a1- and a2-line beams, the line width is $W_2 + W_1$.

3. Results and discussion

In our numerical simulations, the NA of the illumination objective is set to 0.65 with radius equal to $1.0 \times 10^{-3}$ m. The annulus mask at the rear pupil plane corresponds to an inner NA of 0.57 and an outer NA of 0.65. The line beams are parallel to the $z$-axis and locate along the line $k_x = \pm 0.6k_0$ (NA = 0.6) in the pupil plane. In section 2, we have theoretically obtained a fast method to determine the ratio of $L_2$ to $L_1$, as well as $W_2$ to $W_1$, to suppress the side lobes. The a1-line beams are cropped by the outer radius of the annulus and the resulting $L_1 = 9.62 \times 10^{-4}$ m. According to Eq. (10), we have $L_2 = 5.77 \times 10^{-4}$ m. The width of a1-line beams is set as $W_1 = 1.25 \times 10^{-5}$ m and that of a2-line beams $W_2 =2.08 \times 10^{-5}$ m ($W_2/W_1 = 5/3$). The a3-line beams are the combination of a1-line and a2-line beams, as shown in Fig. 2(c). Using the straightforward diffraction integral approach [33], simulations for the light sheets produced by a1-line beams, a2-line beams, and a3-line beams have been performed through Matlab code, respectively. Simulation results of the optical field intensity distribution in $xz$ and $yz$ planes near the focus are shown in Fig. 2. Homogeneous illumination profiles which simulate the experimental dithering effect are also plotted. In order to make an intuitive comparison of the side lobe ratios of different light sheets, all the illumination profiles are normalized. One can see that the relative side lobe level of the light sheet generated by the a3-line beams (Fig. 2(c)) is much lower compared to that by the a1-line beams (Fig. 2(a)) and a2-line beams (Fig. 2(b)). It is worth noting that, compared to that of DLLS 1, such a significant suppression of side lobes is achieved with a slight broadening of the main lobe but without sacrificing the propagation length of the light sheet.

 

Fig. 2. Simulation results of double-line light sheets. Columns, from left to right: the intensity pattern at the rear pupil of the illumination objective; the $xz$ cross-sectional intensity pattern of light sheet at the front focal plane of the objective; $z$-intensity profile of the dithered light sheet; the $yz$ cross-sectional intensity pattern of light sheet. Lines length: $L_1 = 0.98 \times 10^{-3}$ m, $L_2 = 5.77 \times 10^{-4}$ m; Lines width: $W_1 = 1.25 \times 10^{-5}$ m, $W_2/W_1 = 1.66$. The rear pupil illumination restricts to an NA range of 0.57 to 0.65.

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To quantify the side lobe suppression effect, we calculate the ratio of the power of all the side lobes ($P_{\rm {side}}$) and the power of the main lobe ($P_{\rm {main}}$), i.e., $P_{\rm {side}}/P_{\rm {main}}$. The optical power can be calculated by integrating the optical field intensity of the dithered light sheet in the $z$-direction. $I(z)$ is the normalized intensity distribution in the $z$-direction of the dithered light sheet obtained from the simulation. Thus, for a light sheet with known $I(z)$, the total power of the light sheet in the range of $Z_{\rm {min}}$ to $Z_{\rm {max}}$ can be given by:

To check the efficiency of the previously discussed theory of side lobe suppression by superposition of two light sheets, more numerical simulations were performed to investigate the optimization of geometrical parameters of $L_1$ and $L_2$ to obtain the best side lobe suppression effect. To find the optimal ratio of $L_2$ to $L_1$, we fix the length of a1-line beams as $L_1 = 9.62 \times 10^{-4}$ m and vary the length of a2-line beams $L_2$. Figure 3(a) shows $P_{\rm {side}}/P_{\rm {main}}$ and the main lobe FWHM of the light sheet 3 as functions of a2-line beams length $L_2$. In the region where $L_2/L_1$ is between 0.2 and 0.8, $P_{\rm {side}}/P_{\rm {main}}$ is less than 11.2%. Notably, the side lobe elimination is most effective within the range $L_2/L_1 \rightarrow (0.5 \mbox {-} 0.6)$ where $P_{\rm {side}}/P_{\rm {main}}$ is close to 2.0%. This result is in good accordance with the light sheet superposition theory demonstrated in section 2 with $L_2/L_1 = 3/5$ and confirms the main points of the theory presented in this paper. When $L_2/L_1$ is 0.54, $P_{\rm {side}}/P_{\rm {main}}$ of DLLS 3 is 1.97%, which exhibits a 81.3% decrease compared to that of DLLS 1 ($P_{\rm {side}}/P_{\rm {main}}$ of DLLS 1 is 10.3%). It also can be noticed from the FWHM curve in Fig. 3(a) that the reduction of side lobes comes at a cost to the thickness of the light sheet: the main lobe FWHM of DLLS 3 is $\thicksim 1.78\lambda$ which exhibits a 22% increase compared to that of DLLS 1 (which is $\thicksim 1.42 \lambda$).

 

Fig. 3. (a) For fixed $L_1 = 9.62 \times 10^{-4}$ m, $W_1 = 1.25 \times 10^{-5}$ m, $W_2 =2.08 \times 10^{-5}$m, the dependences of $P_{\rm {side}}/P_{\rm {main}}$ (red) and the main lobe FWHM (blue) on the ratio $L_2/L_1$. $P_{\rm {side}}/P_{\rm {main}}$ represents the ratio of all side lobe power to the main lobe power, and the main lobe FWHM represents the thickness of the light sheet. (b) For fixed $L_1 = 9.62 \times 10^{-4}$ m, $L_2/L_1 = 0.60$, $W_1 = 1.25 \times 10^{-5}$ m, the dependences of $P_{\rm {side}}/P_{\rm {main}}$ (red) and the main lobe FWHM (blue) on the ratio $W_2/W_1$. The dots are simulation data while the solid lines are fitting results.

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Another factor influencing the side lobe suppression effect is the width ratio of $W_1$ to $W_2$. Now, we keep the a1-line width constant ($W_1=1.25 \times 10^{-5}$ m) and vary the a2-line width $W_2$ to find the optimal ratio $P_{\rm {side}}/P_{\rm {main}}$. Figure 3(b) shows the $P_{\rm {side}}/P_{\rm {main}}$ and main lobe FWHM of light sheet 3 in dependence of $W_2/W_1$. The $P_{\rm {side}}/P_{\rm {main}}$ firstly decreases and then increases as the $W_2$ increases. In the region where $W_2/W_1$ is between 1.3 and 2.0, the power in the side lobes is further reduced. The minimum is at $W_2/W_1 = 1.66$, where $P_{\rm {side}}/P_{\rm {main}}$ reaches 1.7%. The main lobe FWHM of DLLS 3 increases with $W_2/W_1$. The propagation length (FWHM of the y-profile) and illumination uniformity of DLLS 3 also vary with $L_2/L_1$ and $W_2/W_1$, as shown in Fig. S1 in the supplementary section.

Figure 4 compares the characteristics of the three different light sheets in the $z$-direction. The simulation results with parameters obtained both from the theoretical model and from numerical simulation optimization are plotted. According to the optimization performances in Fig. 3, the parameters $L_2/L_1 = 0.53$, $W_2/W_1 = 1.66$ are taken for the lowest ratio of $P_{\rm {side}}/P_{\rm {main}}$. It can be seen in Fig. 4(c) that the simulation results of light sheet 3 using theoretical parameters and optimization parameters are in good agreement. The intensity profiles of the three different light sheets obtained with theoretical parameters are presented in the same figure for comparison (Fig. 4(d)). It can be clearly seen that, compared to DLLS 1, DLLS 3 has much lower side lobes with a slightly broader main lobe.

 

Fig. 4. (a) – (c) The $z$-intensity profiles of the dithered light sheet DLLS 1, 2 and 3. The dots show the results of simulations performed with parameters from theoretical derivation, while the solid lines show that with parameters from simulation optimization. (d) Comparison of the $z$-intensity profiles of the three light sheets DLLS 1, 2 and 3 in the case using parameters from theoretical derivation.

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The above simulation results confirmed that our method using the principle of superposition of two different light sheets is an efficient and reliable approach to suppressing the side lobes of double-line light sheet. Compared with prior techniques for suppressed side lobes [29], the proposed method has the advantages of direct minimizing the side lobes through easy design, no influence on the propagation length, and very small sacrifice of the main lobe thickness. Furthermore, the proposed method also applies to other kinds of structured light sheets. In the following, we extend this method to FLLS and LBS cases.

As demonstrated in Fig. 5, when moving the double-line beams inward, the double-line beams become clipped by the inner annulus of the mask, which leads to a four-line beam illumination pattern. The corresponding structured light sheet, i.e., a FLLS has decreased main lobe FWHM but increased sidelobe-to-mainlobe energy ratio when compared to the DLLS. A FLLS is a better choice when higher axial resolution is required [21,34]. However, the side lobes of FLLS can approach 90.4% strength of the main lobe and are very hard to remove computationally [35]. Here, we demonstrate the side lobe suppression by the superposition of two different FLLSs. The rear pupil illumination restricts to an NA range of 0.20 to 0.45. The four-line beams are parallel to the $z$-axis and locate along the line $k_x = \pm 0.15k_0$ in the pupil plane. The function of four-line beams distribution can be expressed as: $f_0(z') = rect(\frac {z' + z_c}{L}) + rect(\frac {z' - z_c}{L})$. Here, $L$ is the length of each line beamlet, and $z_c$ is the distance from the center points of the line beamlets to the $x$-axis. Taking the Fourier transform of this function, the amplitude profile of the optical field distribution along $z$-direction in the focal plane is $f_{2F}(z)=\frac {2L}{i (\lambda F)^2}sinc(\pi L \frac {z}{\lambda F})cos(2 \pi \frac {z_c}{\lambda F}z)$. According to the principle of by superposition of two light sheets with different side lobe distributions, the second side lobe peak of light sheet 1 (A point) should be coincident to the first side lobe peak of light sheet 2 (B point). The positions of A and B point can be derived by $z_A=\frac {\lambda F}{z_{c1}}$ and $z_B = \frac {1}{2} \times \frac {\lambda F}{z_{c2}}$, respectively ($z_{c1}$ and $z_{c2}$ are the distances from the center points of the line beamlets to the $x$-axis in FLLS 1 and 2, respectively). When $z_A =z_B$, we have $z_{c1}=2z_{c2}$. It can be noted that $sinc(\pi L \frac {z}{\lambda F})$ determines the side lobe envelope of a FLLS. Hence $L$ should be as large as possible to achieve the lowest possible side lobes. For FLLS 1, the length of each line beamlet $L_1$ just equals the length of the line clipped by the annulus, and for FLLS 2, $L_2$ equals 2($z_{c2} \mbox {-} z_{le}$), where $z_{le}$ is the position of the inner endpoint of each line beam clipped by the annulus. The width of line beams for the FLLS 2 equals $W_2 = (W_1L_1)/L_2$.

 

Fig. 5. Simulation results of three different four-line light sheets. Rows, from top to bottom: FLLS 1; FLLS 2; FLLS 3. Columns, from left to right: the intensity pattern at the rear pupil of the illumination objective; the $xz$ cross-sectional intensity pattern of light sheet at the front focal plane of the objective; $z$-intensity profile of the dithered light sheet; the $yz$ cross-sectional intensity pattern of light sheet. The rear pupil illumination restricts to an NA range of 0.20 to 0.45.

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Figure 5 shows the simulation results corresponding to the three different FLLSs with theoretically derived parameters as demonstrated above. The theoretically derived parameters are $z_{c1} = 3.61 \times 10^{-4}$ m, $z_{c2} = 1.80 \times 10^{-4}$ m, $L_1 = 4.02 \times 10^{-4}$ m, $L_2 = 4.14 \times 10^{-5}$ m, $W_1 = 1.25 \times 10^{-5}$ m, and $W_2 = 1.21 \times 10^{-4}$ m. With these parameters, the simulation results show that, compared to FLLS 1, the ratio of $P_{side}/P_{main}$ of FLLS 3 slightly increases while the first side lobe peak of the FLLS 3 is greatly reduced (by 45.6%). It means that the first side lobe is successfully suppressed, even though the total energy of the side lobes of FLLS 3 fails to be suppressed. This is a consequence of the theory we use to design. In our theory, the first side lobes of light sheets 1 and 2 are designed to interfere strongly and destructively, while other side lobes may not. We definite $P_{1stSide}$ as relative energy of the first side lobe of FLLS 3 (normalized by that of FLLS 1). Figure S2(c) in the supplementary section demonstrates $P_{1stSdie}$ as a function of $z_{c2}/z_{c1}$, and there is a minimum around $z_{c2}/z_{c1} = 0.5$, which is in accordance with our theory. The suppression of the first side lobe can also benefit the improvement of the axial resolution. This is because the first side lobes of FLLS have a greater influence on the axial resolution of the imaging system.

Figure 6 displays the comparison of these three light sheets’ characteristics in the $z$-direction, with both theoretically derived and optimization parameters. The optimization process is shown in Fig. S2 and S3, and for the lowest ratio of $P_{\rm {side}}/P_{\rm {main}}$, the optimization parameters are $z_{c1} = 3.61 \times 10^{-4}$ m, $z_{c2} = 2.6 \times 10^{-4}$ m, $L_1 = 4.02 \times 10^{-4}$ m, $L_2 = 2.01 \times 10^{-4}$ m, $W_1 = 1.25 \times 10^{-5}$ m, and $W_2 = 2.5 \times 10^{-5}$ m. The simulation results of FLLS 3 with these optimization parameters have lowest sidelobe-to-mainlobe energy ratio but show stronger first side lobes which may lead to worse axial resolution. The influence of line beam geometry on the propagation length (FWHM of the y-profile) and illumination uniformity of FLLS 3 are also shown in Fig. S2 and S3.

 

Fig. 6. The $z$-intensity profiles of the dithered light sheet FLLS 1, 2 and 3. The dots show the results of simulations performed with parameters from theoretical derivation, while the solid lines show that with parameters from simulation optimization. (d) Comparison of the $z$-intensity profiles of the three light sheets FLLS 1, 2 and 3 in the case using parameters from theoretical derivation.

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Compared with DLLS and FLLS, the LBS has a higher ratio of side lobes energy, producing more out-of-focus background. However, it is worth noting that, in the propagation direction, the illumination of LBS is uniform, which to some extent, improves the efficiency of light energy utilization. No dithered process is needed, which substantially reduces the complexity of the system.

For the side lobe suppression of LBS, the parameter derivation process is the same as that of FLLS. The rear pupil illumination restricts to an NA range of 0.07 to 0.35. The line beams are parallel to the $z$-axis and locate along the line $k_x = 0$ in the pupil plane. The theoretically derived parameters $z_{c1} = 2.60 \times 10^{-4}$ m, $z_{c2} = 1.30 \times 10^{-4}$ m, $L_1 = 3.55 \times 10^{-4}$ m, $L_2 = 9.54 \times 10^{-5}$ m, $W_1 = 1.25 \times 10^{-5}$ m, and $W_2 = 4.65 \times 10^{-5}$ m are used in the simulations. The simulation results of three different LBSs are illustrated in Fig. 7. We can observe the first side lobes of LBS 3 are significantly suppressed. Figure 8 compares the optical field profile of these three LBSs in the $z$-direction. Simulation results with both theoretically derived and optimization parameters are shown in Fig. 8. The optimization process is shown in Fig. S4 and S5, and for the lowest ratio of $P_{\rm {side}}/P_{\rm {main}}$, the optimization parameters are $z_{c1} = 2.59 \times 10^{-4}$ m, $z_{c2} = 0.87 \times 10^{-4}$ m, $L_1 = 3.55 \times 10^{-4}$ m, $L_2 = 0.09 \times 10^{-4}$ m, $W_1 = 1.25 \times 10^{-5}$ m, and $W_2 = 4.81 \times 10^{-4}$ m are used in the optimization procedure. The optimization results of the LBS 3 also show stronger first side lobe which may cause a worse axial resolution. The influence of line beam geometry on the propagation length (FWHM of the y-profile) and illumination uniformity of LBS 3 are also shown in Fig. S4 and S5.

 

Fig. 7. Simulation results of three different LBSs. Rows, from top to bottom: LBS 1; LBS 2; LBS 3. Columns, from left to right: the intensity pattern at the rear pupil of the illumination objective; the $xz$ cross-sectional intensity pattern of LBS at the front focal plane of the objective; $z$-intensity profile of the light sheet; the $yz$ cross-sectional intensity pattern of light sheet. The rear pupil illumination restricts to an NA range of 0.07 to 0.35.

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Fig. 8. (a)-(c) The $z$-intensity profiles of the light sheet LBS 1, 2 and 3. The dots show the results of simulations performed with parameters from theoretical derivation, while the solid lines show that with parameters from simulation optimization. (d) Comparison of the $z$-intensity profiles of the three light sheets LBS 1, 2 and 3 in the case using parameters from theoretical derivation.

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It should be noted that DR method is also used to suppress the side lobes of the LBS [29]. In DR method, two interfering light sheets were produced via a pupil mask composed of two concentric rings (the so-called double-ring design). With a DR design, the side lobes of the resulting light sheet may be canceled without sacrificing the light sheet thickness. However, a shorter propagation length of the resulting light sheet can be observed because rings in the mask with discrete NA ranges can lead to a field modulation in the propagation direction. In our method, two interfering light sheets are produced through a same ring and hence the propagation length won’t become shorter. Instead, a slight sacrifice of light sheet thickness is observed.

4. Conclusion

In summary, we demonstrated that the side lobes of structured light sheet such as DLLS, FLLS and LBS can be effectively suppressed by superposition of two corresponding light sheets with different side lobe distributions. The resulting new light sheet illuminations provide a much decreased sidelobe-to-mainlobe energy ratio without affecting the propagation length. A theoretical method to quickly determine the illumination line patterns in the rear pupil plane of the illumination objective is proposed, and the effectivity of the method is confirmed by optimization simulations. Our proposed light sheets provide a simple and fast realization for improved imaging contrast without increased risk of sample photodamage. As such, our method paves the way to rapid, in-vivo high-contrast volume imaging applications.