Axial Resolution Enhancement of Optical Sectioning Structured Illumination Microscopy Based on Three-Beam Interference

Abstract

As a branch of 3D microscopy, optical sectioning structured illumination microscopy (OS-SIM) has the advantages of fast imaging speed, weak photobleaching and phototoxicity, and flexible and compatible configuration. Although the method of using the one-dimensional periodic fringe pattern projected on the sample can remove the out-of-focus background from the in-focus signal, the axial resolution of the final reconstructed 3D image is not improved. Here, we propose a three-beam interference OS-SIM, namely TBOS, instead of the common-used dual-beam interference OS-SIM (DBOS). The three-beam interference scheme has been adopted in 3D super-resolution SIM (3D-SR-SIM), where the fringe phase shifting needs to be along each of the three orientations. In contrast, TBOS applies phase shifting only in one arbitrary direction. We built a TBOS SIM microscope and performed the 3D imaging experiments with 46 nm diameter fluorescent microspheres and a mouse kidney section. The axial resolution of the 3D image obtained with TBOS was enhanced by a factor of 1.36 compared to the DBOS method, consistent with the theoretical analysis and simulation. The OS-SIM with enhanced axial resolution for 3D imaging may find a wide range of applications in the biomedical field.

1. Introduction

Optical microscopes are ubiquitous in biomedical research for observing cell and tissue structures and their dynamic events [1,2]. However, due to the limited depth-of-field (DOF) of the objective lens, the in-focus target and out-of-focus background components of the samples are mixed in the recorded image [3], which leads to a low contrast and low signal-to-background ratio in the image, further restricting the ability of 3D imaging of optical microscopy [4,5]. The key to overcoming this drawback of 3D microscopic imaging is to extract the in-focus information from the out-of-focus background to obtain optical sectioning images. The laser scanning confocal microscopy (LSCM) [6,7], two-photon microscopy (TPM) [8,9], light sheet fluorescence microscopy (LSFM) [10,11], and optical sectioning structured illumination microscopy (OS-SIM) [12,13] are the mostly often used methods for this goal.

The pointwise scanning scheme of optical sectioning imaging adopted by LSCM and TPM intrinsically has slow imaging speed [14]. The LSFM, with the alias of selective or single plane illumination microscopy (SPIM) [15], employs two orthogonally arranged objective lenses to excite and detect transparent samples [16,17], which is elegant but complex and expensive to use. In contrast, the OS-SIM is built on a conventional wide-field microscope by replacing the Koehler illumination with a fringe-structured illumination, making the OS-SIM more accessible and affordable.

The principle of OS-SIM is based on the dependence of the projected fringe modulation on the depth of field, that is, the illuminating fringe pattern only has a high modulation depth in the in-focus plane while it decays rapidly in the out-of-focus region [18]. Through three rounds of equilateral phase shifting of the fringe, and the Mean-Square-Root (RMS) reconstruction algorithm [12], the out-of-focus background and the fringe imposed on the in-focus component can be removed, resulting in a sectioned image of the sample. With axial scanning of the sample, a sequentially sectioned image stack can be obtained and a 3D rendering object can be visualized using software. Though OS-SIM can remove the out-of-focus background, it does not improve the axial resolution in the final 3D-reconstructed image. Three-dimensional super-resolution SIM (3D-SR-SIM) enables the improvement in both axial and lateral resolutions with 3D-structured illumination via three- or four-beam interference [19,20]. However, it generally requires 15 steps of phase shifting, i.e., 5-step phase shifting along each of the three fringe orientations, resulting in a high data capacity and low imaging speed.

In this paper, we propose the three-beam interference scheme for optical sectioning SIM, namely TBOS. It owns the optical sectioning capability with enhanced axial resolution. The axial resolution is improved by 1.36 times compared to the common-used dual-beam interference OS-SIM (named DBOS). The phase-shifting step is only one third of the 3D-SR-SIM, benefiting from its phase-shifting operation only along one spatial orientation. The feasibility and merits of the TBOS method were verified with 3D imaging experiments on 46 nm diameter fluorescent microspheres and a mouse kidney section.

2. Principle of TBOS

Assuming a sample labeled with fluorophores has a spatial density distribution of S(r, z), where r indicates the lateral coordination (x, y) and z represents the axial coordination, the intensity distribution of the structured illumination on the sample is I(r, z). The 3D image D(r, z) can be expressed in the form:

$$D(\mathit{r},z)=[I(\mathit{r},z)S(\mathit{r},z)]\otimes h(\mathit{r},z),$$

(1)

where h(r, z) represents the 3D point spread function (PSF) of the system, and the symbol ⊗ denotes the convolution operation. To generate a 3D-structured illumination pattern, three coherent beams are employed for interference. For simplicity, one beam’s wave vector is indicated by k₀ and propagates along the z-axis, and the other two beams (wave vectors denoted by k₁ and k₋₁, respectively) symmetrically transmit along the z-axis at angles of ±θ to the xz-plane, as illustrated in Figure 1a. The amplitudes of the three beams are set to identical. Figure 1b shows the 3D intensity distribution of the formed structured illumination with the three-beam interference.

We further concentrate on the 3D-structured illumination intensity distribution in the xz-plane. The conclusion of xz-plane imaging is accessible to the yz- and xy-plane. The intensity distribution of structured illumination in the xz-plane is described by:

$$I(x,z)=1+\frac{2}{3}\cos (2\pi v_{1}x+2\phi )+\frac{4}{3}\cos (\pi v_{1}x+\phi )\cos (2\pi v_{2}z),$$

(2)

here, $v_1=\frac{1}{{\mathrm{T}}_1}=\frac{2n\sin \theta }{\lambda }$ and $v_2=\frac{1}{{\mathrm{T}}_2}=\frac{n(1-\cos \theta )}{\lambda }$ denote the spatial frequencies of the interference pattern in the lateral and axial directions, respectively. n is the refractive index of the surrounding medium, λ represents the wavelength of emission light, φ = φ₀ + mΔφ denotes the phase of the fringe, φ₀ denotes the initial phase, and Δφ represents the phase step between consecutive images; m is the order of the phase shifting.

Considering the depth-of-focus of the objective lens, the sample distribution S(x, z) can be divided into two components of the in-focus portion S_in(x, z) and the out-of-focus background S_out(x, z). Under the structured illumination, only the in-focus S_in(x, z) is modulated by the illumination pattern I(x, z). Substituting Equation (2) into Equation (1), we obtain:

$$\begin{gathered} D\left(x,z\right)=[S_{in}\left(x,z\right)+S_{out}\left(x,z\right)]\otimes h\left(x,z\right)+\frac{2}{3}\left[\cos \left(2\pi v_{1}x+2\phi \right)S_{in}\left(x,z\right)\right]\otimes h\left(x,z\right)+\frac{4}{3}[\cos (\pi v_{1}x+ \\ \phi )\cos \left(2\pi v_{2}z\right)S_{in}\left(x,z\right)]\otimes h\left(x,z\right). \end{gathered}$$

(3)

For the image acquired with a 2D camera, the coordinate z is kept fixed, so, the term of $\cos (2\pi v_{2}z)$ can be regarded as a constant. Therefore, Equation (3) can be rewritten as:

$$\begin{gathered} D\left(x,z\right)=[S_{in}\left(x,z\right)+S_{out}\left(x,z\right)]\otimes h\left(x,z\right)+\frac{2}{3}\left[\cos \left(2\pi v_{1}x+2\phi \right)S_{in}\left(x,z\right)\right]\otimes h\left(x,z\right) \\ +\frac{4}{3}[\cos \left(\pi v_{1}x+\phi \right)S_{in}\left(x,z\right)]\otimes H\left(x,z\right), \end{gathered}$$

(4)

where H(x, z) is defined as an equivalent PSF, $H\left(x,z\right)=h\left(x,z\right)\cos \left(2\pi v_{2}z\right)$. In Equation (4), the first term represents the wide-field image, composed of the in-focus information S_in(x, z) and the out-of-focus background S_out(x, z). The second and third terms imply the sectioned image, which only includes the S_in(x, z), but noting the PSF h(x, z) of the third convolution is replaced by H(x, z) modulated by $\cos \left(2\pi v_{2}z\right)$.

The equivalent PSF H(x, z) is determined by the parameters of both the optical system and the illumination pattern, which is consequently compressed due to the cosine modulation. To prove this quantitatively, we use the system’s theoretical PSF $h\left(x,z\right)={\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{c}}^2\left[kn{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\alpha /2)(x,z)\right]$ [21] in H(x, z). Here, k = 2π/λ, n denotes the surrounding refractive index, and α represents the aperture angle of the objective lens. The $\cos \left(2\pi v_{2}z\right)$ is modulated by the axial frequency v₂, which has a relationship with the lateral frequency v₁:

$$v_2=\frac{n\left\{1-\cos \left[\arcsin (\frac{\lambda }{2n}{v}_1)\right]\right\}}{\lambda }.$$

(5)

To maintain consistency with the afterward experimental conditions, we select the wavelength λ = 560 nm, and an oil-immersion objective lens of 60×/NA1.49 matching n = 1.515. The dashed curve in Figure 2a presents the system’s axial PSF h(x, z), where z = 0 indicates the axial location of the transverse in-focus plane. The equivalent PSFs H(x, z) corresponding to different v₁ are also plotted in Figure 2a. It can be seen that the full width at half maximum (FWHM) of H(x, z) is smaller than that of h(x, z), which means the axial resolution is improved. However, the intensity of the side lobes of H(x, z) gradually increases with the increase of v₁, which will affect the final reconstructed image quality and even lead to artifacts. According to the previous research [18] that indicates a maximal optical sectioning strength obtained around v₁ = 0.5, as shown in Figure 2b, we select v₁ = 0.6 to obtain the optimized axial resolution enhancement and minimized influence of the side lobes in practice. In this case, the FWHMs of h(x, z) and H(x, z) are 0.677 μm and 0.483 μm, respectively, corresponding to an axial resolution improved by 1.4 times.

To obtain the sectioned image with improved axial resolution, in Equation (4), we set:

$$\left\{\begin{array}{c}{g}_w=[S_{in}\left(x,z\right)+S_{out}\left(x,z\right)]\otimes h\left(x,z\right)\\ g_c=\frac{2}{3}[\cos \left(2\pi v_{1}x+2{\phi }_0\right)S_{in}\left(x,z\right)]\otimes h\left(x,z\right)\\ g_s=-\frac{2}{3}[\sin \left(2\pi v_{1}x+2{\phi }_0\right)S_{in}\left(x,z\right)]\otimes h\left(x,z\right)\\ q_c=\frac{4}{3}[\cos \left(\pi v_{1}x+{\phi }_0\right)S_{in}(x,z)]\otimes H\left(x,z\right)\\ q_s=-\frac{4}{3}[\sin \left(\pi v_{1}x+{\phi }_0\right)S_{in}(x,z)]\otimes H\left(x,z\right)\end{array}\right..$$

(6)

Then, Equation (4) becomes a simplified form:

$$D(x,z)=g_w+g_c\cos (2m\mathsf{\Delta }\phi )+g_s\sin (2m\mathsf{\Delta }\phi )+q_c\cos (m\mathsf{\Delta }\phi )+q_s\sin \left(m\mathsf{\Delta }\phi \right).$$

(7)

To solve the five unknown parameters of $g_w$, $g_c$, $g_s$, $q_c$, and $q_s$, it requires five equations, which can be created by setting the phase difference Δφ$=\frac{2}{5}\pi$ and the order of phase shifting m = 0, 1, 2, 3, 4. Then, we obtain the following equation group:

$$\left[\begin{array}{c}{D}_0(x,z)\\ D_1(x,z)\\ D_2(x,z)\\ D_3(x,z)\\ D_4(x,z)\end{array}\right]=\left[\begin{array}{ccccc}1& 1& 0& 1& 0\\ 1& \cos \frac{4}{5}\pi & \sin \frac{4}{5}\pi & \mathrm{c}\mathrm{o}\mathrm{s}\frac{2}{5}\pi & \sin \frac{2}{5}\pi \\ 1& \cos \frac{8}{5}\pi & \sin \frac{8}{5}\pi & \mathrm{c}\mathrm{o}\mathrm{s}\frac{4}{5}\pi & \sin \frac{4}{5}\pi \\ 1& \cos \frac{12}{5}\pi & \sin \frac{12}{5}\pi & \mathrm{c}\mathrm{o}\mathrm{s}\frac{6}{5}\pi & \sin \frac{6}{5}\pi \\ 1& \cos \frac{16}{5}\pi & \sin \frac{16}{5}\pi & \mathrm{c}\mathrm{o}\mathrm{s}\frac{8}{5}\pi & \sin \frac{8}{5}\pi \end{array}\right]\left[\begin{array}{c}{g}_w\\ g_c\\ g_s\\ q_c\\ q_s\end{array}\right].$$

(8)

By solving Equation (8), we obtain $g_w$, $g_c$, $g_s$, $q_c$, and $q_s$, and then we can calculate the optical sectioned image for TBOS with the following equation:

$$D_{\mathrm{T}\mathrm{B}\mathrm{O}\mathrm{S}}\left(x,z\right)=\sqrt{{q_c}^2+{q_s}^2}.$$

(9)

It is noticed that this formula is similar to the amplitude demodulation algorithm for the two-beam optical sectioning SIM [22,23], that is $\propto \sqrt{{g_c}^2+{g_s}^2}$. However, here we obtain an improved axial resolution OS image with Equation (9) due to the compressed PSF H(x, z).

3. Simulation Results

Numerical simulation is performed at first to validate the superiority of TBOS, than DBOS, in axial resolution. The standard USAF1951 resolution chart is used as the simulated object, as shown in Figure 3a. For DBOS, three-step phase shifting with an interval of 2π/3 is conducted, and the optical sectioning image is reconstructed according to the conventional RMS algorithm, shown in Figure 3c. In addition, the wide-field image can also be obtained by summing up the three raw images, shown in Figure 3b. For TBOS, five-step phase shifting with an interval of 2π/5 is conducted, and the optical sectioning image is reconstructed according to Equations (8) and (9), shown in Figure 3d. It is clear that the TBOS image is much sharper than the DBOS image. Elements 9–6 corresponding to a spatial frequency of 912.3 lp/mm in the TBOS image can be clearly resolved, while they are blurred for the DBOS image. The normalized intensity profiles in Figure 3i along the same cut-lines of Figure 3e–h support the axial resolution improvement of TBOS over the DBOS.

We also compare the lateral resolving powers of TBOS and DBOS in the xy-plane. Similarly, the USAF1951 resolution chart is used as the target, as shown in Figure 4a. The reconstructed wide-field image, DBOS image, and TBOS image are shown in Figure 4b–d, respectively. It can be seen that the TBOS image and the DBOS image have a similar resolving power, while the imaging contrast of the DBOS is a little bit higher than the TBOS, seen from the zoomed regions (elements 11–3 and 11–4) in Figure 4e–h and the normalized intensity profiles in Figure 4i. The reason is because the lateral frequency of the fringe is v₁/2 in the case of TBOS, which is half of that of the DBOS (v₁ = 0.6), therefore causing a lower optical sectioning capability than the DBOS in the lateral direction. Nevertheless, the difference is not remarkable as seen from the optical sectioning strength curve of Figure 2b.

4. Experimental Results and Discussion

4.1. Experimental Setup

The TBOS SIM microscope was built to verify the theory and simulation results. The optical layout of the setup is illustrated in Figure 5. A laser beam (λ = 473 nm) with horizontally linear polarization (GEM473, Laser Quantum Inc., Stockport, UK) is expanded by Lens 1 (f₁ = 10 mm) and Lens 2 (f₂ = 250 mm), and then illuminates a ferroelectric liquid crystal spatial light modulator (FLC-SLM; QXGA-3DM, 2048 × 1536 pixels, Forth Dimension Displays Inc., Dalgety Bay, UK) through a polarizing beam splitter (PBS) and a half-wave plate (HWP). The FLC-SLM can rapidly generate and shift the diffractive gratings displayed on it, dividing the laser beam into multiple diffraction beams. The 0th and ±1st order diffraction beams are selected by a spatial filter in the focal plane of Lens 3 (f₃ = 500 mm), and then relayed by a 4f system consisting of Lens 4 (f₄ = 175 mm) and Lens 5 (f₅ = 125 mm) to the back focal plane of the objective lens (Apo TIRF, 60×/NA1.49, Nikon Inc., Tokyo, Japan). The 3D-structured illumination pattern is generated in the focal region of the objective lens via the three-beam interference. The sample is mounted on an assembly motorized XY and piezo Z-axis translation stage (XY, PZ-2000FT; Z, PZ-2500FT, Applied Scientific Instrumentation Inc., Eugene, OR, USA). The fluorescence signal is filtered by a dichroic beamsplitter (Di03-R488/561, Semrock Inc., Rochester, NY, USA) and an emission filter (FF01-523/610-25, Semrock Inc., Rochester, NY, USA), and then collected by a tube lens (f = 300 mm) and imaged on the sCMOS camera (C11440-22CU, 2048 × 2048 pixels, maximum 100fps, Hamamatsu Inc., Hamamatsu, Japan). For comparison, the TBOS SIM microscope can be easily changed to the DBOS SIM configuration by blocking the 0th-order diffraction beam.

4.2. Axial Resolution Measurement

The 46 nm diameter fluorescence bead slide (TransFluoSpheres, carboxylate-modified, Ex.488 nm/Em.560 nm, Thermo Fisher Inc., Waltham, MA, USA) is used to measure the axial resolutions of the TBOS and DBOS systems. The lateral frequency of the fringe is measured to be 3367 lp/mm, and the system’s lateral cut-off frequency is 2NA/λ = 2 × 1.49/(560 × 10⁻⁶) = 5321 lp/mm, thus v₁ ≈ 0.63. The sample is axially scanned in a range of 4.2 μm with a step of 100 nm.

The maximum intensity projection (MIP) images in the xy- and xz-plane of the sectioned image stacks for the wide-field, DBOS, and TBOS methods are shown in Figure 6a–c, respectively. We select some sub-volumes (as marked 1, 2, 3) to zoom-in for the detailed comparison, as seen in Figure 6d–i, respectively. To obtain the quantitative value of the lateral and axial resolution of the system, we select 50 beads randomly in the field to measure their lateral and axial FWHMs as shown in Figure 6j,k. After Gaussian fitting and statistics, the average lateral FWHMs for the wide-field, DBOS, and TBOS are 326 ± 21 nm, 303 ± 30 nm, and 288 ± 25 nm, respectively. The average axial FWHMs for those methods are 897 ± 51 nm, 895 ± 46 nm, and 660 ± 47 nm, respectively. The measurements indicate that the DBOS method does not improve the lateral and axial resolution compared to the wide-field microscopy, but the TBOS method does improve the axial resolution by 1.36 times compared with the DBOS and the wide-field microscopy. Through this calibration measurement with fluorescent beads, it is verified that our proposed TBOS method has superiority over the conventional DBOS method in axial resolution, despite the lateral resolution not being improved, implying the 3D imaging capability of the TBOS still has space to be improved.

4.3. Imaging Result of Biological Sample

After the resolution calibration with fluorescent beads, a biological specimen of a tri-chrome-labelled mouse kidney section slide (FluoCells prepared slide #3, labelled with Alexa Fluor^® 488 WGA, Ex.358 nm/Em.461 nm, Alexa Fluor^® 568 phalloidin, Ex.495 nm/Em.519 nm, and DAPI, Ex.578 nm/Em.600 nm, Thermo Fisher Inc., Waltham, MA, USA) is used to test the imaging merit of the TBOS method. The sample is axially scanned in a range of 12.2 μm with a step of 200 nm, generating 62-slice-sectioned images. The MIP images in the xy- and xz-plane of the stacked image for the wide-field, DBOS, and TBOS methods are shown in the upper and middle rows of Figure 7a–c, respectively. The single cross-section OS images in the xz-plane along the white cut-lines in the upper section are also shown in the lower rows of Figure 7a–c, respectively. It is obvious that the TBOS and DBOS images are sharper than the wide-field image, proving that both TBOS and DBOS have the background removal ability. Further, we select some sub-volumes in the specimen (as marked 1, 2, 3) to zoom-in for detailed comparison, as seen in Figure 7d–i, respectively. The TBOS has a better axial resolving power of discrimination of the fine structure of the sample than the DBOS method, which can be seen from the zoomed-in MIP images in the xz-plane of Figure 7h,i, and it is more apparent in the normalized intensity profiles in Figure 7j. The 3D volume visualization of the sample with the different methods is shown in Figure 7k–m, respectively. Through this experiment, it further demonstrates that compared with the wide-field microscopy, both the TBOS and the DBOS methods can remove the out-of-focus background to obtain the optical sectioning image, but the TBOS can further improve the axial resolution rather than the DBOS. Although the influence of the side lobes of PSF on the final reconstructed image quality is the main factor of limiting the 3D imaging capability of TBOS, it is worthy to investigate the way to eliminate the side lobes to further improve the 3D imaging capability.

5. Conclusions

In this work, we have proposed a three-beam interference OS-SIM (TBOS). The theoretical analysis and numerical simulation demonstrated that the TBOS method can improve the axial resolution of the 3D optical section image by 1.4 times compared with the dual-beam interference OS-SIM (DBOS). The experiments with 46 nm diameter fluorescent microspheres and a mouse kidney section further proved that the TBOS method obtained a 1.36-fold improvement in the axial resolution compared to the DBOS method. Though the three-beam interference scheme has been adopted in 3D-SR-SIM, the TBOS method applies phase shifting only in one direction, reducing the raw image amount by a three-fold extent and thus offering a faster 3D imaging speed. We expect the TBOS method to be found in more applications in the biomedical field.