3.1.

Influence of Distance Parameter α and Initial Phase Difference δ

To achieve optimal ILS effects, we systematically investigated the influence of interference parameters on the uniformity of light intensity and the beam cross-sectional profile. Initially, we focused on the uniform distribution of the interference beam along the propagation axis. We defined the uniformity of the interference beam by the intensity ratio $R$ ($\alpha ,\delta$), which represents the ratio of the intensity at the origin ($r_s=0,y_s=0$) to the intensity at the focal point ($r_s=0,y_s=\mathrm{\Delta }$):

To determine the appropriate distance parameter $\alpha$, we considered the conditions for interference enhancement, where $\cos (\mathrm{\Phi })$ equals 1 in Eq. (9), corresponding to the origin ($r_s=0$, $y_s=0$). We plotted the variation of $R$ with $\alpha$, as shown in Fig. 2(a). When $\alpha$ is 0, the two beams overlap, resulting in $R=1$. As $\alpha$ increases, $R$ initially rises, reaching its maximum value at $\alpha =0.56$, before decreasing. For $0<\alpha <1$, the intensity at $y_s=0$ is greater than at $y_s=\mathrm{\Delta }$. This indicates that for smaller $\alpha$ values, the intensity profile along the $y_s$ axis forms a single peak. With increasing $\alpha$, the profile gradually splits into two peaks, resulting in a decrease in intensity between the peaks, as illustrated in Fig. 2(b). At $\alpha =2.3$, which corresponds to a separation of 4.6 times the Rayleigh length between the two beams, the intensity at $y_s=0$, $r_s=0$ is 48.4% of the peak intensity, approximately half, as depicted by the blue solid line in Fig. 2(b).

Fig. 2

Influence of parameters $\alpha$ and initial phase difference $\delta$. (a) Variation of the intensity ratio between $y_s=0$ and the focal point with respect to $\alpha$. (b) Intensity curve along the $y_s$ axis. (c) The intensity variation at $y_s=0$ as a function of the radial coordinate $r_s$ and $\delta$. In this plot, the radial coordinate $r_s$ is normalized by $w_0$, and the intensity is normalized by the maximum intensity. (d) The relationship curve between the intensity $I$ ($r_s=0$, $y_s=0$) and $\delta$. The green curve represents $I$ ($r_s=0$, $y_s=0$) after being normalized by the maximum value, and the red curve represents the uniformity curve $R(\delta )$. (e) The variation of the beam cross-sectional intensity at different $\delta$ values, specifically $\delta =1.23$, 4.39, and 5.2. (f) The relationship between the first zero of the intensity $I$ $(r_s,y_s=0)$ and the variation of $\delta$. (g) Intensity $I(z_s)$ at $y_s=0$ under different $\delta$ conditions, normalized to $w_0$ coordinates. (h) Variation of intensity along the propagation axis, solid line represents beam intensity, dashed line represents intensity after scanning the beam along the $x_s$ axis, normalized to Rayleigh length $Y_r$ coordinates. In (e), the scale is $2w_0$.

To enhance the uniformity of light intensity by varying the initial phase ($\delta$), we explore the impact of $\delta$ while keeping the distance parameter $\alpha$ constant. Figure 2(d) presents the intensity and uniformity at $y_s=0$, $r_s=0$ as a function of $\delta$, with the intensity values normalized to their maximum. As $\delta$ increases from 0 to $2\pi$, the intensity initially decreases (reaching a minimum at $\delta \approx 1.23$), then gradually rises to its maximum (around $\delta \approx 4.39$), followed by a decrease, exhibiting a cosine-shaped curve, as indicated by the green curve in Fig. 2(d). Correspondingly, the uniformity first decreases, then increases, and finally decreases again, with a minimum value of $R(\delta =1.23)=0$ and a maximum value of $R(\delta =5.2)=56\%$. It is worth noting that the maximum values for both intensity and uniformity correspond to slightly shifted $\delta$ values. This discrepancy arises because changing $\delta$ not only affects the intensity at the origin but also impacts the intensity at other locations in space, as evident from the combined use of Eqs. (9) and (12). To further comprehend the influence of $\delta$ on the cross-sectional intensity, we plotted the radial light distribution at $y_s=0$ as a function of $\delta$ in Fig. 2(c). Around $\delta =1.23$, destructive interference occurs at the origin, and as $\delta$ increases, the energy at the center ($r_s=0$) of the beam gradually strengthens. Near $\delta =4.39$, interference enhancement occurs, resulting in maximum intensity. Figure 2(e) illustrates the intensity distributions at $y_s=0$ for $\delta$ values of 1.23, 4.39, and 5.2. The corresponding intensity profiles along the $z_s$ axis are shown in Fig. 2(g). At $\delta =1.23$, destructive interference leads to the formation of a hollow beam [Fig. 2(e), red line in Fig. 2(g)]. Increasing $\delta$ causes the energy to converge toward the center, with maximum interference intensity occurring near $\delta =4.39$. As $\delta$ continues to increase, the intensity at the origin starts to decrease, but the uniformity remains in an ascending phase [Fig. 2(d)]. The maximum uniformity is achieved at $\delta =5.2$. Figure 2(h) provides a comparison of three scenarios: destructive interference ($\delta =1.23$), interference enhancement ($\delta =4.39$), and maximum uniformity ($\delta =5.2$) for $\alpha =2.3$. It displays the intensity distribution along the propagation axis ($y_s$) and the intensity distribution of the LS formed by scanning along the $x_s$ axis, with the intensity normalized to the maximum value. When destructive interference occurs, the intensity at the origin is zero, resulting in the poorest uniformity along the axial direction. At $\delta =5.2$, both the uniformity of the beam intensity and the uniformity of the scanned LS (82.2%) are slightly superior to the scenario at $\delta =4.39$ [indicated by the blue curve in Fig. 2(h)]. When the LS is formed by scanning along the $x_s$ axis, we observe that the intensity becomes more uniform along the $y_s$ axis, as depicted by the dashed line in Fig. 2(h). This is because in the scanning mode, the intensity along the $x_s$ axis is integrated, and thus the sidelobes contribute to improving the uniformity of the LS. In Fig. 2(h), we also observed a phenomenon where the maximum intensity varies with $\delta$, similar to the focal shift phenomenon reported in previous studies.³¹

In addition to intensity uniformity, the size of the beam spot is equally important to consider. Using Eq. (10), we can plot the curve of the first zero point ($r_{s0}$) of the intensity distribution at the $y_s=0$ plane as a function of $\delta$ [Fig. 2(f)]. When $\delta$ is 0, $r_{s0}$ is $\sim 1.3w_0$. As $\delta$ gradually increases from 0 to 1.23, the intensity of the central lobe decreases, and the zero point moves closer to the center, resulting in a decrease in the first zero point as $\delta$ increases. Around $\delta =1.23$, complete destructive interference occurs, causing the central lobe to disappear, and the first zero point becomes $r_{s0}=0$. As $\delta$ further increases, the central intensity remains nonzero, and the first zero point becomes the first zero point outside the central ring, leading to a sudden change in the curve as shown in Fig. 2(f). Subsequently, $r_{s0}$ decreases as $\delta$ increases, following a pattern consistent with Fig. 2(c). When $\delta$ is 5.2, $r_{s0}$ is $1.78w_0$. At this point, when the intensity of the main lobe drops to $1/e^2$ of its maximum, the corresponding $r_s$ is $1.386w_0$, which is smaller than the beam radius at the Rayleigh length of a Gaussian beam ($\surd 2w_0$).

Based on the analysis above, it is evident that interference beams exhibit side lobes, and there is an energy transfer between the main lobe and side lobes as the value of $\delta$ changes. Therefore, it is necessary to investigate the relationship between the main lobe, side lobes, and the value of $\delta$ in interference beams.

According to Eq. (7), the interference intensity is the product of ($4I_{\mathrm{\Delta }}$) and the cosine squared term. The first term is resembling the noninterfering beam at $y_s=0$, and the second term is representing the interference modulation. Figures 3(a)–3(d) illustrate the first term, second term, and interference intensity for $\alpha =2.3$ at different $\delta$ values (1.23, 3.55, 4.39, and 5.2), with intensity normalization. When the two main peaks in the second term are widely separated, the interference intensity forms a hollow spot with a zero main lobe and the strongest side lobes [Fig. 3(a)]. As the two main peaks approach each other, the annular spot converges inward. Around $\delta =3.55$, the interference intensity resembles a flat-top beam with side lobes peaking at $\sim 6.5\%$ of the main lobe [Fig. 3(b)]. At $\delta =4.39$, when the two peaks merge into one, the interference enhancement reaches its peak [Fig. 3(c)], with side lobes peaking at around 7.8% of the main lobe. Increasing $\delta$ further narrows the central peak of the second term, resulting in a thinner main lobe and a slight decrease in peak intensity while the side lobe intensity increases. At $\delta =5.2$, the side lobes peak at $\sim 13\%$ of the main lobe peak. Figure 3(e) presents the intensity distribution in the $y_s=0$ plane for various $\delta$ values, clearly demonstrating the evolution of the intensity profile with varying $\delta$. Additionally, Fig. 3(f) shows the trend of the PRSM as a function of $\delta$. When $\delta$ is $<0.31$, the main lobe is stronger than the side lobes. In the range of $0.31<\delta <0.9$, the side lobe energy increases rapidly. Within the range of $0.9<\delta <1.63$, the energy of the first side lobe significantly surpasses that of the central lobe, resulting in a hollow-like field. As $\delta$ ranges from 2.7 to 3.55, the PRSM approaches 1, resulting in a flat-top field. Importantly, when $3.55<\delta <5.2$, the main lobe predominates, with the first side lobe being very small ($<0.13$), and the second side lobe nearly absent [Figs. 3(c) and 3(d)]. This characteristic arises due to the exponential decay of side lobes as they are further away from the main lobe, distinguishing interference beams significantly from Bessel beam. As we all know, Bessel beam have multiple side lobes: the first side lobe typically carries around 16% of the main lobe’s peak intensity, followed by the second side lobe with $\sim 9\%$, and the third side lobe with about 8%… Each side lobe in a Bessel beam carries an equal amount of energy.³² In contrast, interference beams generally have only one side lobe, and the ratio of the side lobe’s peak intensity to the main lobe’s peak intensity can vary depending on the specific interference pattern and parameters.

Fig. 3

Modulation effect of $\delta$ on the beam profile. (a)–(d) The beam profiles at different $\delta$ values, where the red dashed line represents the first term of the interference intensity, the green dashed line represents the second term of the interference intensity, and the blue solid line represents the total interference intensity. The intensity values have been normalized. (e) The distribution of intensity at different $\delta$ values on the $z_s$, $x_s$ plane. (f) The peak intensity ratio between the side lobes and the central lobe. (PRSM: peak ratio of the first side lobe to the main lobe). The scale in (e) is $2w_0$.

In summary, we conducted a comprehensive study on the effects of distance parameter $\alpha$ and initial phase difference $\delta$ on the uniformity of light intensity, intensity distribution, and spot size. We observed that the uniformity of light intensity initially increases and then decreases as $\alpha$ increases. To expand the FOV, we selected two light beams separated by 4.6 Rayleigh lengths and investigated the influence of different $\delta$ values on the intensity distribution. Remarkably, when $\delta$ is set to 5.2, the intensity uniformity on the $r_s=0$ axis of the scanning plate reaches 82.2%. When $\delta$ is 5.2, the radius of the main lobe ($1/e^2$) is $1.386w_0$ ($<\surd 2w_0$). Additionally, we observed that the main lobe size ($r_{s0}$) of the spot decreases as $\delta$ increases, indicating that larger $\delta$ values contribute to smaller spot sizes. The modulation of interference terms resulted in variations in the beam profile with changing $\delta$, including the emergence of hollow structures, flat distributions, or dominance of the main lobe. In comparison to Bessel beam, interference beams exhibit a significantly smaller number of side lobes due to exponential decay as the distance increases, highlighting a distinct advantage of interference beams.

3.2.

Comparative Analysis of Three Beam Profiles: Uniformity of Intensity and Beam Width

To evaluate the performance of LSs formation using three different beams, we conducted simulations with the following parameters: illumination aperture $\mathrm{NA}=0.16$, $\alpha =2.3$, $\lambda =0.6328\mathrm{nm}$, $\delta =5.2$; detection objective aperture ${\mathrm{NA}}_d=0.5$. Figures 4(a)–4(f) present the intensity distribution of the $y_s-z_s$ cross section and the intensity profile at $y_s=0$ for the three beams and their corresponding scanned LSs. The Gaussian beam focuses at the origin ($y_s=0$), whereas the two noninterfering beams and the interfering beam have their foci at a distance $\mathrm{\Delta }=2.3Y_r$ from the origin ($y_s=0$). The Gaussian beam has the smallest spot size at the origin [Fig. 4(a)]. The noninterfering beam has a spot size of $\sim 2.5w_0$ at the origin, as shown in Fig. 4(b), the blue curve in Fig. 4(h), and its intensity rapidly decreases along the $y_s$ axis to 30.6% of the peak value [Fig. 4(g), the blue solid line]. However, when interference occurs, the beam size decreases [Fig. 4(c)], and the intensity at the origin increases to 53.7%, which is 1.75 times higher than the noninterfering case. The ILS achieves a uniformity of 82% [Fig. 4(g), dashed line], thus demonstrating better uniformity than the NILS. To assess the beam width, we define the beam width as the radial radius at which the intensity drops to $1/e^2$ of the peak value in the cross section. As shown in Fig. 4(h), the Gaussian beam has the smallest beam width, $w_0$, at the origin, and the beam width at a distance $Y_r$ from the origin is $\surd 2w_0$. Similar to the Gaussian beam, for both interference beams and noninterference beams, we can define the interval in which the beam width is less than the $\surd 2w_0$ as the effective length, representing the FOV of the LS. Therefore, the FOV for the GLS is $2Y_r$. If the two beams do not interfere, their intensities simply add up, resulting in a wider beam and an increased beam width of $2.5w_0$ at the origin. The effective length with a beam width smaller than $\surd 2w_0$ is only $3.12Yr$ [Fig. 4(h), solid blue line]. This means that noninterfering superposition leads to a degradation in beam quality because the two noninterfering beams rapidly diverge when moving away from their respective foci, and the diffused light overlaps with the other beam, resulting in an increased beam width. However, for interference beam, the beam width significantly decreases, and at $y_s=0$, the beam width is $\sim 1.386w_0$, smaller than $\surd 2w_0$ [Fig 4(h), solid red line]. The length with a beam width smaller than $\surd 2w_0$ is $\sim 5.88Y_r$ for interference beam. The intensity curves at $y_s=0$ are shown in Fig. 4(i), where the intensity profile of the interference beam is significantly smaller than that of the noninterference case due to the modulation by the cosine squared term. It should be noted that the interference beam exhibits side lobes; however, under the aforementioned parameters, the ratio of the first side lobe to the main peak is 0.13, smaller than $1/e^2$. Although the side lobes of the scanned beam formation lead to an undesired increased background intensity, with the background peak accounting for $\sim 0.2$ of the main peak, the full width at half maximum (FWHM) of the ILS ($\mathrm{FWHM}=2w_0$) is reduced by 37.5% compared to the NILS ($\mathrm{FWHM}=3.2w_0$). The system point spread function of the LSFM is represented as the product of the detection point spread function and the illumination point spread function. As shown in Fig. 4(j), the point spread function of the ILS is smaller than that of the NILS case, and the interfering side lobes only result in a minor background intensity.

Fig. 4

Comparison of three types of beams and LSs. (a)–(c) The intensity distributions of Gaussian beam, noninterference beam, and interference beam on the $y_s$, $z_s$ plane. (d)–(f) The intensity profiles of GLS, NILS, and ILS on the $y_s$, $z_s$ plane. The corresponding subplots on the right-hand side of (a)–(f) show the intensity distributions of the beams in the $y_s=0$ plane. (g) The intensity curves along the propagation axis ($y_s$) for the three types of beams and their corresponding LS. (h) The variation of beam width along the propagation axis for Gaussian beam, noninterference beam, and interference beam. (i) The intensity variation of $I(z_s)$ at $y_s=0$ for the three types of beams and their corresponding LS. (j) The system point spread functions for the three types of LS. The parameters used are $\mathrm{NA}=0.16$, $\alpha =2.3$, $\lambda =0.6328\mathrm{nm}$, and ${\mathrm{NA}}_d=0.5$. The white dashed line indicates the focal plane where $y_s=0\mu \mathrm{m}$. The scale in (a)–(f) is $20\mu \mathrm{m}$.

In summary, we systematically investigated the influence of the distance parameter $\alpha$ and the initial phase parameter $\delta$ on the interfering light field. Then under the condition of $\alpha =2.3$, we optimized the initial phase difference to enhance the uniformity of the light intensity, resulting in an improved uniformity of 82.2% for the ILS. By utilizing the interference effect to modulate the beam profile, the thickness of the interfering LS was reduced by 37.5% compared to the NILS, effectively expanding the FOV of the LS to nearly three times that of the GLS.

4.1.

Sample Preparation

We have prepared two types of samples, namely a quantum dot fluorescent solution and fluorescent microspheres, in order to compare the performance of GLS, NILS, and ILS. The quantum dots used in our experiment are water-soluble cadmium-based quantum dots, with a concentration of $10\mathrm{nm}/\mathrm{ml}$ and a photoluminescence full-width at half-maximum (PL/FWHM) of 671/40 nm. To observe the trajectory of the light beam, we introduce $\sim 10\mu \mathrm{l}$ of the quantum dot solution into a rectangular sample chamber filled with water (dimensions: $21.9\mathrm{mm}\times 63.3\mathrm{mm}\times 27\mathrm{mm}$). As for the fluorescent microspheres, we utilize $0.2\mu \mathrm{m}$ silica fluorescent microspheres (632/680 nm). These microspheres are embedded in low-melting-point agarose at a ratio of 1:240 for the experiment.

4.2.

Experimental Setup

To assess the performance of the ILS, we construct a custom experimental setup depicted in Fig. 1(b). We utilize a CW laser source (HNL100LB, 632.8 nm, 10 mw) as the excitation light, with its polarization aligned along the $x_s$ axis. The incident laser is split into red and blue light paths by a BS. In the red path, a retroreflector mirror adjusts the optical path length to control coherence. To regulate the relative focal distances of the two beams at the objective’s back focal plane (Olympus UPlanSApo 4×/0.16 NA), we employ an SLM (HOLOEYE, PLUTO-2) loaded with a defocus phase. The SLM is conjugated with the galvanometer mirrors and the objective’s entrance pupil. The other beam is directed into the common optical path through BS2. During the experiment, the blue light path is obstructed when imaging with the GLS. For ILS imaging, we adjust the mirror to ensure interference between the two beams. In the case of NILS imaging, a half-wave plate is inserted into the blue light path and rotated to render the light’s polarization perpendicular to the $x_s$ axis. Meanwhile, the optical path length of the red path is extended to prevent interference between the two beams. The sample is positioned on a high-precision motorized translation stage (NanoFaktur, SFS-D10600) capable of $z$ axis movement. To capture fluorescence, we employ a water-immersion objective with a numerical aperture of 0.5 (Olympus UMPlanFl 20× Water Immersion, 0.5 NA) as the detection objective, and a tube lens (ETL = 200 mm) is positioned behind the detection objective. The detection objective lens is mounted on a movable translation stage, aligning the focal plane with the plane of LS. Fluorescence images are recorded by an sCMOS (ORCA-Flash BT C15440, Hamamatsu) camera at a rate of 50 frames per second. A filter (680 nm/FWHM 50 nm) is inserted in front of the camera to eliminate excitation light and stray light. The experiment involves imaging the trajectories of the optical paths and fluorescent microspheres.

5.1.

Experimental Results of Beam Fluorescence Trajectories

The fluorescence trajectories of the beam in the quantum dot solution are shown in Fig. 5. For the Gaussian beam, the beam width is the narrowest only at the center ($y_s=0$) of the FOV, rapidly diverging away from the center, as illustrated in Fig. 5(a). If the spacing between two noninterfering beams focusing is greater than the Rayleigh length, the beam at the intermediate position of the focal point is noticeably thicker, as shown by the blue line in Figs. 5(b) and 5(d). When the two beams can interfere, the interference effect significantly reduces the beam width, as depicted in Figs. 5(c) and 5(d). Compared to noninterfering beams, the interference effect between the two beams results in a more uniform beam width in the FOV, as shown in Figs. 5(b) and 5(c). The interference beam width is it reduced by a factor of 1.647 compared to the noninterfering beam. For the interference beam, the width of the beam at the sides of the FOV are smaller, with the thickest part at the center, $\sim 1.416$ times the minimum beam width of the Gaussian beam, as shown in Fig. 5(c). Compared to noninterference beams, interference beams exhibit the capability of reducing beam width and creating a flatter beam width curve. As shown in Fig. 5(e), for Gaussian beam, the effective FOV, where the beam width is less than the √2 times the Gaussian beam min waist width (Min ${\text{Width}}_{\text{Gaussian}}$), is $\sim 270\text{pixels}$ ($75.6\mu \mathrm{m}$). In contrast, for noninterference beams, the effective FOV, where the beam width is less than √2 Min ${\text{Width}}_{\text{Gaussian}}$, is 579 pixels ($162.1\mu \mathrm{m}$). However, for interference beams, the effective FOV is 880 pixels ($246.4\mu \mathrm{m}$). Interference expands the FOV by $\sim 1.5$ times compared to the noninterference situation. Therefore, compared to the NILS, the ILS has a smaller thickness and a longer effective length.

Fig. 5

Experimental results of beam fluorescence trajectories in water-soluble quantum dot solution. (a) Gaussian beam fluorescence trajectory; (b) noninterfering beam fluorescence trajectory; (c) interfering beam fluorescence trajectory; and (d) beam fluorescence intensity distribution, corresponding to the white dashed line in the center ($y_s=0$) of the FOV. (e) The beam width variation curves along the propagation axis of Gaussian beams, noninterference beams, and interference beams. The black dashed line in the image represents the square root of 2 times the minimum beam waist width of the Gaussian beam. The yellow dashed lines represent the FOV boundaries on both sides, in (a)–(c), the yellow dashed line in (a) corresponds to the green dashed line in (e), whereas the yellow line in (b) and (c) corresponds to the red dashed line in (e). Scale: $63\mu \mathrm{m}$.

5.2.

Experimental Results of Fluorescent Microsphere Imaging

To further validate the performance of ILS, we conducted three-dimensional imaging of $0.2\mu \mathrm{m}$ silica fluorescent microspheres embedded in low-melting agarose blocks using GLS, NILS, and ILS as excitation LS. The imaging volume was $674\times 674\times 40\mu \mathrm{m}$ with a $z$ axis step size of $0.1\mu \mathrm{m}$. The imaging results are shown in Fig. 6, where the yellow dashed line represents the FOV boundary of the scanning LS.

Fig. 6

Three-dimensional imaging results of $0.2\mu \mathrm{m}$ fluorescent microspheres. (a) Microsphere imaging results illuminated by GLS. (b) Three-dimensional imaging results of microspheres illuminated by ILS. (c) Three-dimensional imaging results of microspheres illuminated by NILS. Panels (d)–(f) correspond to $y_s{z}_s$ cross sections in (a)–(c), respectively. (i)–(iii) The fluorescence intensity curves of microspheres numbered 1, 2, and 3 in (d) along the $z$ direction. (iv)–(vi) The fluorescence intensity curves of microspheres numbered 4, 5, and 6 in (e) along the $z$ direction. (vii)–(ix) The fluorescence intensity curves of microspheres numbered 7, 8, and 9 in (f) along the $z$ direction. The scale in (a)–(c) is $100\mu \mathrm{m}$, whereas the scale in (d)–(f) is $36\mu \mathrm{m}$. The frame exposure time is 20 ms.

First, for GLS illumination, the fluorescent microspheres in the center of the FOV appeared bright, whereas those farther from the center quickly became dim, as shown in Fig. 6(a). This observation is consistent with the intensity distribution characteristics of GLS along the propagation axis. In the case of ILS illumination, the region of the fluorescent microspheres illuminated in the FOV was noticeably larger, and the fluorescence intensity of the microspheres was more uniform, as shown in Fig. 6(b). However, for NILS, the fluorescence brightness of the microspheres near the center of the FOV ($y_s=0$) was significantly darker compared to the microspheres at the focal point of the scanning beam, as shown in Fig. 6(c). This is consistent with the results in Figs. 5(b) and 5(e), indicating that NILS have a larger thickness in the center, resulting in lower energy density.

To quantitatively analyze the axial resolution of the three types of LSs, we selected $y_s{z}_s$ cross sections from the three-dimensional volumes of the microspheres imaged by each LS [Figs. 6(d)–6(f)]. We chose microspheres close to $y_s=0$ and microspheres on both sides of the FOV, and then performed Gaussian fitting on their intensity along the $z_s$ axis to obtain their $z_s$ axis resolution [Figs. 6(i)–6(ix)].

First, taking the imaging results of fluorescent microspheres illuminated by GLS as an example, the $y_s{z}_s$ cross section is shown in Fig. 6(d). We selected microsphere 2 in the center of the FOV and microspheres 1 and 3 on both sides (close to the Rayleigh length). Then we plotted the intensity distribution curves of these three microspheres along the $z_s$ axis and performed Gaussian fitting, as shown in Figs. 6(i)–6(iii). The $1/e^2$ radius of microsphere 2 was $1.97\mu \mathrm{m}$, microsphere 1 was $2.71\mu \mathrm{m}$, and microsphere 3 was $2.71\mu \mathrm{m}$. The calculated ratio of $z_s$ axis resolution differences was $2.71/1.97=1.376$.

Next, taking the imaging results of fluorescent microspheres illuminated by ILS as an example, the $y_s{z}_s$ cross section is shown in Fig. 6(e). We selected three microspheres, microsphere 5 in the center of the FOV, and microspheres 4 and 6 on both sides, and plotted their intensity curves along the $z_s$ axis, as shown in Figs. 6(iv)–6(vi). The $1/e^2$ radii of microspheres 4, 5, and 6 were 2.2, 2.55, and $2.19\mu \mathrm{m}$, respectively. The calculated ratio of $Z$ axis resolution differences was $2.55/2.19=1.164$.

Finally, taking the imaging results of fluorescent microspheres illuminated by NILS as an example, the $y_s{z}_s$ cross section is shown in Fig. 6(f). We selected three microspheres in the FOV and plotted their intensity distribution curves along the $z_s$ axis, as shown in Figs. 6(vii)–6(ix). The $z_s$ axis resolutions of microspheres 7, 8, and 9 were 2.19, 3.85, and $2.3\mu \mathrm{m}$, respectively. The calculated ratio of $z_s$ axis resolution differences was $3.85/2.19=1.758$.

Based on the data presented in Table 1, we are able to observe the axial resolution of three types of LSs along the $z_s$ axis. By combining our previous analysis with the findings from this table, we can draw the following conclusions: ILS exhibits a larger FOV compared to the other two LS types. Additionally, ILS demonstrates a more uniform axial resolution within the FOV. This implies that whether in the central or peripheral regions, we can expect consistent image quality and resolution when using ILS. Therefore, ILS demonstrates advantages in terms of both FOV and axial resolution.

Table 1

Axial resolution of three types of LSs in the left, center, and right FOV.