Single-molecule localization microscopy (SMLM)^[1–4] has pushed the resolution to sub-40 nm, making it a powerful superresolution imaging modality that reveals the organisms of subcellular structures. Through maximum likelihood estimation (MLE)^[5,6], SMLM can precisely determine the coordinates of each sparsely distributed blinking fluorophore. However, due to the finite photon number in a single blinking event and the significant background brought by high power density excitation, the experimental lateral localization precision is confined to around 20 nm, which can be derived mathematically by Cramér–Rao lower bound (CRLB)^[7–10]. Several three-dimensional (3D) SMLM methods have also been reported in the last decade^[11–17]. Thanks to point spread function (PSF) engineering and adaptive optics (AO), 3D-SMLM breaks the symmetry of PSF along the axial direction, making axial superresolution possible. However, these techniques blur the PSFs laterally, leading to a deterioration of lateral resolution. A 4Pi microscope^[18–21] is also a feasible solution and achieves ultrahigh axial resolution to even sub-10 nm. Nonetheless, considering the instability of fluorescence interference and the complexity of the detection path^[22], this scheme has not been widely used.

To improve the 3D localization precision theoretically and empirically, structured illumination has been introduced in SMLM in recent research. For example, by utilizing interference sinusoidal fringes with phase shifting in the transverse plane, SIMFLUX^[23] and repetitive optical selective exposure (ROSE)^[24] achieved a lateral resolution improvement of around twofold in the orthogonal directions of $x$ and $y$. ModLoc^[25] and ZIMFLUX^[26] introduced an oblique excitation beam with the epi-illumination to form oblique interference patterns in objective space, achieving sub-10 nm axial localization precision. It should be noted that, due to the distinctive illumination method, the angle between the two excitation beams in the objective space was relatively small, making it difficult to control the pitch of the oblique stripes within a small range, limiting the magnitude of resolution improvement. The advent of ROSE-Z^[27] did achieve the theoretically narrowest axial patterns. However, its lateral resolution has not been further improved.

From what was mentioned above, it can be observed that the recent modulation-enhanced SMLM method can achieve resolution improvement either laterally or axially, which means localization precision can be improved in only one direction, and isotropic spatial resolution is difficult. Here, we report isoFLUX, a method that synergizes PSF engineering and oblique sinusoidal illumination patterns by employing two opposite-placed excitation objective lenses. By fully utilizing the large excitation angle of high numerical aperture (NA) objective lenses, isoFLUX can generate ideal fringes with narrow pitch in both lateral and axial projections. The oblique patterns are generated through synchronous control of the electro-optic modulators to change the direction and phase of two excitation beams. By performing three phase shifts on the patterns tilted in the $x–z$ and $y–z$ directions (six phase shifts in total), isoFLUX can simultaneously improve the lateral and axial localization precision. Combined with PSF engineering, the lateral and axial resolution improvement can be extended to a deeper defocus range, resulting in the optimal 3D resolution. In our simulations, isoFLUX achieves an unprecedented twofold localization precision improvement in both lateral and axial directions within a $\pm 600\mathrm{nm}$ axial range compared to the 3D-SMLM method using the same PSF engineering strategies. We believe this method can open a promising avenue for ultrahigh superresolution microscopy and help reveal the mysteries of the subcellular structures at the molecular level. The simulation code and data set for isoFLUX are made open source and can be found in Ref. [35].

In isoFLUX imaging, the 3D sinusoidal illumination pattern is generated through the interference of excitation lights emanating from two opposite-positioned objective lenses (Fig. 1). Compared to the illumination scheme utilizing a single objective lens, interference illumination by opposing objective lenses exhibits larger transverse and axial momenta (smaller transverse and axial pitches), significantly improving lateral and axial localization precision.

The intensity field of the interference pattern can be described as $$I(x,z)=I_0\{1+m_x\times \sin [{\overrightarrow{k}}_{xz}·(x,z)+\varphi ]\},$$(1)where ${\overrightarrow{k}}_{xz}=[\frac{2\pi }{\lambda }(\cos {\theta }_{x_1}+\cos {\theta }_{x_2}),\frac{2\pi }{\lambda }(\sin {\theta }_{x_1}+\sin {\theta }_{x_2})]$ and $\lambda$ is the wavelength in the sample medium. $m_x$ denotes the modulation depth of the sinusoidal pattern. ${\theta }_{x_1}/{\theta }_{x_2}$ is the angle between the excitation beam from the upper/lower objective lens and the transverse plane. For convenience, we set ${\theta }_{x_1}$ equal to ${\theta }_{x_2}$ hereafter.

To ensure isotropic resolution, we phase-shift the pattern equidistantly 3 times with an interval of 120 deg for both $x$ and $y$ orientations, resulting in six subimages in total. We will have $$\{\begin{array}{c}{I}_x^i(\overrightarrow{r})=\frac{N_x}{3}[1+m_x\times \sin ({\overrightarrow{k}}_{xz}·(x,z)+{\varphi }_x+i·\frac{2\pi }{3}\left)\right]\\ I_y^j(\overrightarrow{r})=\frac{N_y}{3}[1+m_y\times \sin ({\overrightarrow{k}}_{yz}·(y,z)+{\varphi }_y+j·\frac{2\pi }{3}\left)\right]\\ i,j=-1,0,1\end{array},$$(2)where $N_x/N_y$ is the total photon number of three subimages along the $x/y$ direction.

Suppose the single molecule is localized at ${\overrightarrow{r}}_0=(x_0,y_0,z_0)$ with a normalized 3D PSF $h(\overrightarrow{r})$, the measurement of photons in the camera pixel $k$ centered at $(x_k,y_k)$ for each of the subimages can be expressed as $$\{\begin{array}{c}{\mu }_x^i[k]={\iint }_{x_k-\frac{w}{2},y_k-\frac{w}{2}}^{x_k+\frac{w}{2},y_k+\frac{w}{2}}{I}_x^i({\overrightarrow{r}}_0)\times h(x-x_0,y-y_0,z_0)\mathrm{d}x\mathrm{d}y+b{g}_x^i\\ \approx I_x^i({\overrightarrow{r}}_0)\times h(x_k-x_0,y_k-y_0,z_0)\times w^2+b{g}_x^i\\ {\mu }_y^j[k]={\iint }_{x_k-\frac{w}{2},y_k-\frac{w}{2}}^{x_k+\frac{w}{2},y_k+\frac{w}{2}}{I}_y^j({\overrightarrow{r}}_0)\times h(x-x_0,y-y_0,z_0)\mathrm{d}x\mathrm{d}y+b{g}_y^j\\ \approx I_y^j({\overrightarrow{r}}_0)\times h(x_k-x_0,y_k-y_0,z_0)\times w^2+b{g}_y^j\\ i,j=-1,0,1\\ k=1,2,\cdots ,M\end{array},$$(3)where $bg$ denotes the background photons per pixel and $w$ is the pixel size of the detector. The normalization of the PSF intensity is to ensure the total photon number along the $x/y$ direction equals $N_x/N_y$.

The theoretical localization precision of isoFLUX can be calculated by Fisher information matrix $F$. Suppose we are interested in the uncertainty of $\theta =[N_x,N_y,m_x,m_y,x_0,y_0,z_0,b{g}_x^i,b{g}_y^j]$ ($i,j=-1,0,1$ and 13 parameters in total); thus, $F$ will be a $13\times 13$ square matrix with elements $F_{lm}$ calculated as follows: $$\begin{gathered} F_{lm}=\sum _{k=1}^M(\sum _{i=-1,0,1}\frac{1}{{\mu }_x^i[k]}\frac{\partial {\mu }_x^i[k]}{\partial {\mathit{\theta }}_l}\frac{\partial {\mu }_x^i[k]}{\partial {\mathit{\theta }}_m} \\ +\sum _{j=-1,0,1}\frac{1}{{\mu }_y^j[k]}\frac{\partial {\mu }_y^j[k]}{\partial {\mathit{\theta }}_l}\frac{\partial {\mu }_y^j[k]}{\partial {\mathit{\theta }}_m}). \end{gathered}$$(4)

The optimal estimation precision of the $l$th parameter is given by $${\sigma }_l=\sqrt{{[F^{-1}]}_{ll}},$$(5)where ${\sigma }_l$ is known as the $l$th diagonal element of the CRLB. It is worth noting that it will involve calculating the partial derivative of $h(x-x_0,y-y_0,z_0)$ with respect to $x_0/y_0/z_0$ in Eq. (4). This partial derivative can be easily obtained by applying the cubic spline to the PSF model^[28,29].

One possible setup of isoFLUX is shown in Fig. 1. In the excitation path, isoFLUX utilizes four electro-optic modulators (EOMs) to select the direction of linear polarization (EOM1 in Fig. 1), realize the phase-shifting of interference patterns (EOM2), and change the direction of interference fringes (EOM3 and EOM4). It should be noted that EOMs are not the only solution to achieve interference and phase-shifting of the patterns; a galvanometer-piezo module can be an alternative. However, we believe that EOMs are particularly well suited for isoFLUX due to their ultrafast modulation speed (up to kilohertz) and high optical transmission efficiency. After being modulated by EOMs and relayed by 4f systems (L1, L3 and L2, L5, respectively), the two excitation beams focus on the back focal planes of two objective lenses, respectively (OBJ1 and OBJ2 in Fig. 1), guaranteeing the convergence positions are symmetrical along the common focal point of the two objectives and generating an oblique interference pattern. In the detection path, isoFLUX only uses the lower objective lens (OBJ2 in Fig. 1) to collect the fluorescence, dramatically simplifying the structure of the detection path compared to the 4Pi microscope. Before propagating to a deformable mirror (DM) conjugated to the back focal plane to perform PSF engineering, the fluorescent signals are relayed by a 4f system (L7 and L8) to match the recommended optical aperture of the DM with the pupil plane. After the DM, the fluorescence is collected by an imaging lens (L9) and imaged on an sCMOS camera.

The general localization process of isoFLUX includes PSF calibration, candidate single-molecule detection, parameter estimation of illumination pattern, and single-molecule localization. For PSF calibration, one can obtain the PSF model by in vitro^[6] or in situ methods^[30–32]. However, in situ PSF retrieval considers the aberration introduced by the sample, thus enabling the acquisition of the PSF with better accuracy. Estimating illumination pattern parameters $({\overrightarrow{k}}_{xz},{\overrightarrow{k}}_{yz},{\varphi }_x,{\varphi }_y)$ is also a critical factor in achieving high-precision localization, and methods based on Fourier spectrum estimation can be utilized^[23,26].

As for single-molecule localization, MLE, which takes into account both pattern information and information of 6 sub-images, can achieve the optimal experimental localization precision. The loss function of MLE is given as $$\begin{gathered} {\chi }_{\mathrm{MLE}}^2=2\sum _{i=1}^3[\sum _k({\mu }_x^i[k]-x_k^i)-\sum _k{x}_k^i\ln \left(\frac{{\mu }_x^i[k]}{x_k^i}\right)] \\ +2\sum _{j=1}^3[\sum _k({\mu }_y^j[k]-x_k^j)-\sum _k{x}_k^j\ln \left(\frac{{\mu }_y^j[k]}{x_k^j}\right)]. \end{gathered}$$(6)Here, ${\mu }_x^i[k]$ and ${\mu }_y^j[k]$ are the expected numbers of photons in pixel $k$ of $x$ subimages and $y$ subimages that can be derived from the forward imaging model [Eq. (3)] combined with cubic-spline interpolated PSF model, and $x_k^i$, $x_k^j$ are the corresponding measured numbers of photons. To minimize ${\chi }_{\mathrm{MLE}}^2$, the Levenberg–Marquardt (L-M) algorithm is an ideal choice and is widely used in least-squares fitting for its low time complexity and robustness^{[5,6,8,33,34]}. In subsequent simulation experiments, we also used the L-M based MLE to determine the coordinates of the single molecules.

The theoretical 3D localization precision (CRLB) of isoFLUX and 3D-SMLM is calculated using an astigmatic PSF (60 nm astigmatism aberration was induced) and a saddle-point PSF with 3000 photons/localization (total photon number of 6 subimages) and 5 background photons/pixel (30 background photons/pixel in total). As for parameters of the illumination patterns, the modulation depth in both x and y directions is set as 0.95, the period of oblique pattern is 211 nm, and ${\theta }_{x_1}/{\theta }_{x_2}/{\theta }_{y_1}/{\theta }_{y_2}$ are all set to 40 deg. 2D PSFs from several axial positions of the astigmatic PSF model and the saddle-point PSF are shown in Fig. 2(a) for reference. In the case of using astigmatic PSF, isoFLUX provides ∼twofold average precision improvement over 3D-SMLM in both lateral and axial directions [Fig. 2(c1)]. It can be seen that the lateral and axial localization precision of isoFLUX at each $z$ depth is nearly equal, indicating quasi-isotropic 3D resolution by isoFLUX [Fig. 2(b1)]. Although isoFLUX can provide more significant lateral improvements in defocus positions, the lateral localization precisions are still far inferior to the in-focus position. The issue of degraded precision at defocus $z$ positions can be solved by PSF engineering. We optimize the pupil to obtain optimal 3D localization precision within a specific $z$ range ($-1$ to 1 µm in this simulation) and get a saddle-point PSF, which has been reported before^[17]. Combined isoFLUX with PSF engineering, we get $\sim 1.5$-fold average precision improvement in lateral and over 2.5-fold in axial [Fig. 2(c2)]. The lateral and axial localization precision is almost unchanged with different $z$ depths [Fig. 2(b2)], achieving spatially invariant quasi-isotropic 3D localization precision of less than 5 nm.

To verify the feasibility of isoFLUX, we simulate the line structures and conduct single-molecule Monte Carlo experiments [Figs. 2(d) and 2(e)]. Single-molecule images were generated from the cubic-spline representation of the PSF and multiplied by different photon numbers for each subimage according to the relative position of the single molecule with respect to the sinusoidal patterns. Each subimage was then added with a constant background and finally degraded by Poisson noise. For fitting, isoFLUX is fitted using a self-written MLE algorithm, as described in Sec. 2.4, and 3D-SMLM is fitted by first summing up all six subimages and then using the state-of-art algorithm (SMAP2018, which can be found in Ref. [6]). We fit the width of the reconstructed line structures at each $z$ depth with unimodal Gaussian distribution and give the full width at half-maximum (FWHM) value to represent the axial resolution of the current $z$ position [Figs. 2(d1) and 2(d2)]. It can be seen that isoFLUX significantly improves the axial resolution compared to 3D-SMLM, which aligns well with the CRLB. In Figs. 2(e1) and 2(e2), we conduct repeatability experiments on a single fixed molecule [localized at (0, 0) in lateral with different $z$ depths, repeated 2500 localizations for each depth] and give the scatter plot. The localization precision in the $x$ direction is given aside by calculating the standard deviation of the localization coordinates. Similarly, isoFLUX also shows good improvement in lateral direction compared to 3D-SMLM, which is consistent with the theoretical CRLB.

To validate the performance of isoFLUX, we generate the raw data of a simulated nuclear pore complex (NPC) using the saddle-point PSF model [Fig. 2(a2)] under the same signal-to-noise ratio level and illumination parameters as in Sec. 3, and compare the reconstructed results by isoFLUX and 3D-SMLM (Fig. 3). In the simulation, the fluorophores are distributed from $-600$ to 600 nm, and each polygon is assigned a diameter of 50 nm with a distance between adjacent individual elements set to 19.1 nm. It can be observed that in regions with large out-of-focus depths, the individual structure of the polygons was difficult to resolve using 3D-SMLM [Figs. 3(b1) and 3(b2)]. In contrast, isoFLUX can resolve the structure easily [Figs. 3(a1) and 3(a2)]. We perform bimodal Gaussian fitting on the NPC diameter [Fig. 3(a1)] and get 45.9 and 46 nm for the diameters of two different NPC structures, which is relatively consistent with the ground truth. In addition, we also get the distance of 17.2 nm between adjacent individual elements of a single NPC, close to the ground truth. It is important to note that the axial resolution of isoFLUX also exceeds that of 3D-SMLM [Figs. 3(a3) and 3(b3)]. The axial extension of single NPC structures in isoFLUX is significantly narrower than that of 3D-SMLM.

We further simulate the microtubule data using the saddle-point PSF model [Fig. 2(a2)] and compare the reconstructed results by isoFLUX and 3D-SMLM. The simulation parameters are consistent with the NPC simulation but with 5000 photons/localization. The ground truth of the microtubules is obtained from the EPFL 2016 SMLM Challenge training data set.^[36] The hollow structure of the microtubules was easily distinguishable in isoFLUX [Figs. 4(a1)–4(a3)] but vague in 3D-SMLM [Figs. 4(b1)–4(b3)], indicating an improvement in both lateral and axial resolution. By bimodal Gaussian fitting of the diameters of the microtubules [Figs. 4(a1)–4(a3)], we estimated the diameters of the hollow structures as 15.5, 18.6, and 16.7 nm at three different positions, respectively, which were close to the ground truth (outer diameter was 25 nm with an inner hollow tube of 15 nm diameter, 20 nm for median). Compared to ModLoc, isoFLUX demonstrated simultaneous improvements in both axial and lateral resolution, while ModLoc only reported an improvement in axial resolution.

In summary, we propose isoFLUX as a new modulation-enhanced SMLM modality, enhancing the localization precisions in all three dimensions and achieving spatially invariant isotropic resolution when combined with PSF engineering strategy. isoFLUX is a promising technique for ultrahigh superresolution imaging and will help unravel the intricacies of subcellular structures at the molecular scale. To further push the resolution to a higher level, illumination modes other than sinusoidal patterns can be applied to increase Fisher information content. 4Pi detection can also be considered combined with isoFLUX to further improve the axial resolution, though this inevitably increases system complexity.