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TOMOTRAP optically manipulates and controls an arbitrarily shaped 3D sample by creating optimised 3D trapping light. The trapping light is optimised to resemble the 3D RI distribution of an optically trapped sample, n(r). In this condition, the electromagnetic field energy is maximised, and the optical trapping is most stable according to the electromagnetic variational principle22. To understand the maximisation condition, we assume the scalar diffraction theory in the weak scattering regime. If the scattered field is negligible compared to the incident field, Ein(r). the time-averaged electromagnetic field energy is approximately
$$ U_{{\mathrm {field}}} \approx \frac{{\varepsilon _0}}{2}{\int} {n^2({\bf{r}})|E_{{\mathrm {in}}}({\bf{r}})|} ^2{\mathrm {d}}{\bf{r}} $$ (1) where ε0 is the vacuum permittivity23, 24. The optical trapping is targeted only at a sample. Thus, we subtract the background field energy from the total field energy and maximise the relative field energy as
$$ {\Delta}U_{{\mathrm {field}}} \approx \frac{{\varepsilon _0}}{2}{\int} {{\Delta}n^2({\bf{r}})|E_{{\mathrm {in}}}({\bf{r}})|} ^2{\mathrm {d}}{\bf{r}} $$ (2) where Δn2(r) = n2(r) – nm2, and nm the medium RI. According to the Parseval theorem, the 3D summation of the input field intensity remains constant if the laser power is constant, that is, ${\int} {|E_{{\mathrm {in}}}({\bf{r}})|^2{\mathrm {d}}{\bf{r}}} = C$. With this, the Cauchy–Schwarz inequality dictates that Eq. (2) is maximised when |Ein(r)|2 is proportional to Δn2(r).
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To experimentally demonstrate isotropic microtomography using TOMOTRAP, we combined optical diffraction tomography (ODT) for real-time RI microtomography25 and HOTs for simultaneous optical manipulation of a sample16 (Fig. 1 and the section "Materials and methods"). ODT is one of 3D quantitative phase imaging techniques, which reconstructs the 3D RI distribution of a sample from multiple 2D holograms measured at various illumination angles2, 25 (Fig. 1b). Incident plane waves were scattered by the heterogeneous RI contrast of a sample and recorded as an interferogram. To extract transmitted complex fields from the raw holograms, we used a field retrieval algorithm based on the Fourier transform26. From the retrieved fields, we reconstructed the 3D RI map of the sample using the Fourier diffraction theorem with Rytov approximation27, 28. Note the missing cone problem causes low axial resolution3.
Fig. 1 Experimental setup.
a Optical setup (DMD digital micromirror device, L lens, M mirror, CL condenser lens, OL objective lens, DM dichroic mirror, BS beam splitter, CAM camera). b Optical diffraction tomography (ODT). The sample is illuminated at different incident angles, and the corresponding off-axis holograms are recorded. The raw holograms are converted to transmitted fields using the field-retrieval algorithm. The obtained fields are mapped to the Fourier space, and the inverse Fourier transform reconstructs the refractive index (RI) tomogram with low axial resolution. c Tomographic mould for optical trapping (TOMOTRAP). The phase distribution of the spatial light modulator (SLM) in the Fourier plane was optimised to make the light amplitude maximally overlap the regularised relative sample RI map via a 3D Gerchberg–Saxton (GS) iterative algorithm. d Postprocessing for isotropic 3D reconstruction. A reconstructed tomogram was registered with the initial raw RI tomogram in 3D to estimate the actual 3D orientation. Registration data were applied to obtain a spectrum from a different orientation. Finally, we obtain an isotropically synthesised tomogram and a spectrum devoid of the missing cone.HOTs implemented TOMOTRAP by experimentally generating the desired structured light in a real-time manner based on the measured RI tomogram (Fig. 1c). The target 3D amplitude of the light trap was the contrast of the reconstructed RI tomogram regularised by the nonnegativity constraint. We generated the light trap using a phase-only spatial light modulator (SLM) in the Fourier plane. We optimised the phase pattern of the SLM using a 3D Gerchberg–Saxton (GS) iterative algorithm.
After rotating the optically trapped sample to the desired orientation and recording its image, we reconstructed the isotropic RI tomogram in the postprocessing step (Fig. 1d). From the reconstructed RI tomogram of the optically rotated sample, we estimated the actual change in the orientation of the sample using an iterative 3D registration algorithm (see the "Materials and methods"). For a seamless isotropic reconstruction, we used the estimated registration data to obtain a rotated 3D sample spectrum. Followed by the repeated registration, we selected successful data to obtain a synthesised spectrum without the missing cone and a subsequently isotropic RI tomogram.
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We first validated the feasibility of the proposed method using a standard sample whose structure was simple, and the 3D RI distribution was well known. For this purpose, we tested the optical rotation of a colloidal suspension of 3 μm diameter poly(methyl acrylate) (PMA) bead dimers in a 70% aqueous solution of glycerol (Fig. 2a; see the section "Materials and methods" for sample preparations). To obtain a quasi-spherical spectrum with as small numbers of rotations as possible, we rotated the optically trapped particle to target angles varying from −45° to 90° with an interval of 15° (Fig. 2b). The 3D registration algorithm successfully estimated and corrected the actual 3D orientation of the optically rotated sample (Fig. 2c). The measured misorientation angles with respect to the lateral pitch axis of the sample were within ±4°, with the exception of 90° where the deviation exceeded 12°. We speculate that the highest angle deviation at the angle of 90° resulted as the particle was inclined towards the optical axis, where multiple light scattering is significant. Nevertheless, the misorientation was mitigated by the corrected registration data, and the resultant isotropic RI tomogram was successfully reconstructed (Fig. 2d). In visual inspection, the heterogeneous protrusions on the colloid were clearly reconstructed, which suggests that the reconstruction process can provide an isotropic RI tomogram with high fidelity.
Fig. 2 Validation of the isotropic reconstruction process using a standard sample.
a Raw RI tomogram of a PMA dimer reconstructed by conventional ODT. b, c Sliced tomogram images of the sample rotated by TOMOTRAP. b Raw data. c Registered results. The lateral pitch axis of the raw data was rotated by 45° with respect to the x-axis, and registered to the y-axis. The yellow and white dashed lines indicate rotational angles and the cross-sections, respectively. d Isotropic RI tomogram obtained from the registered data.We examined the isotropic RI reconstruction performance for various colloidal suspensions (Fig. 3). We tested a PMA dimer and a trimer, and successfully rotated them by [−45°, −30°, 30°, 45°, 90°] and [−45°, −30°, 30°, 45°], respectively (see Fig. S1a, b). We compared the RI tomograms that were reconstructed from the conventional ODT without sample rotation with those from our proposed method, which exhibited marked differences. In the conventional ODT with the missing cone problem, both the PMA dimer and trimer exhibited axially elongated artefacts (Fig. 3a, b). In contrast, our proposed method using TOMOTRAP clearly reconstructed the spherical shapes of the 3 μm diameter multimers with improved RI contrasts (Fig. 3c, d). Interestingly, the sample spectra without the missing cone in our method revealed the interferometric signals that are characteristic of multimeric particles29. More importantly, the isotropic RI tomograms allowed enhanced recognition of the heterogeneous surfaces on the beads. These visual results highlight the importance of high-resolution assessment in microtomography.
Fig. 3 3D RI reconstruction of PMA multimers.
a, b RI tomograms and the corresponding scattering potential spectra of a PMA (a) dimer and (b) trimer using the conventional ODT. c, d Corresponding results of the PMA (c) dimer and (d) trimer using our proposed method. (Dashed lines, 2D slice regions; inset, 3D-rendered images). e, f Line plots along the coloured dashed lines for (e) the dimer and (f) trimer (dashed circles, guiding lines for ideal 3 μm diameter spheres with RI of 1.48).To quantitatively analyse the enhanced axial resolution, we compared the axial RI profiles of the reconstructed colloidal particles (Fig. 3e, f). We first plotted the axial RI profiles along the centre of one of the monomers in the PMA dimer (Fig. 3e). In the conventional ODT, the RI profile was blurred along the axial direction and the RI value was underestimated compared with the nominal diameter of 3 μm and RI value of 1.476. On the other hand, the RI profile obtained from our proposed method closely agreed with the expected RI profile based on the nominal sample specifications, with a sharper full width half maximum (FWHM) of 3.3 μm and a higher peak RI value of 1.485. We next analysed the axial RI profiles of a heterogeneous protrusion observed in the PMA trimer (Fig. 3f). Consistent with the previous results, our proposed method improved the axial resolution and RI contrast of the singular structure. The estimated FWHM and peak RI value of the singular structure improved from 1.67 to 1.38 μm and from 1.49 to 1.51, respectively. Thus, we successfully verified the applicability of the proposed method for various multimeric colloidal suspensions.
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Accurate 3D imaging and control of live RBCs are some of the most elusive tasks in microtomography. RBCs are susceptible to the missing cone problem owing to their axially thin biconcave structures. Moreover, they are easily deformed; therefore, rotating them without deformation requires careful sample manipulation. These shortcomings can be mitigated by employing TOMOTRAP.
We isotropically reconstructed the RI of two types of live mouse RBCs suspended in phosphate-buffered saline (PBS): a normal RBC and an echinocyte (Fig. 4; see the section "Materials and methods"). We successfully rotated the normal RBC by 45° and 60°, and the echinocyte by −60°, −45°, −30°, 30°, and 90° using TOMOTRAP, with minimal deformation and precise registration (see Fig. S1c). The effect of removing the missing cones was significant. The missing cone problem in the conventional ODT caused vacant artefacts at the centres of both cells, which are also observed in the simulations30 (Fig. 4a, b). However, our method resolved the biconcave dimples and folded structures of the cells (Fig. 4c, d). Notably, the estimated RI values of both samples reached a maximum of 1.39, without regularisation. The collective results suggest the general feasibility of TOMOTRAP for in situ isotropic microtomography of various freestanding specimens.
Fig. 4 3D RI reconstruction of live mouse red blood cells (RBCs).
a, b RI tomograms and the corresponding scattering potential spectra of (a) a normal RBC and (b) echinocyte using the conventional ODT. c, d Corresponding results of (c) the normal RBC and (d) the echinocyte using our proposed method (dashed lines, 2D slice regions; inset, 3D-rendered images). -
We conclude the analysis with a quantitative comparison of the axial resolution between the conventional ODT and the proposed method. We defined the axial resolution as the FWHM of a 3D coherent spread function (CSF), which is the inverse Fourier transform of the coherent transfer function (the bandwidth range of the 3D Fourier spectrum; Fig. 5). The analysis showed that the isotropic reconstruction of the PMA dimer exhibited the maximum improved axial resolution (Fig. 5a, b). Conventional ODT suffered from an axially elongated artefact in the obtained CSF; however, our proposed method remained unaffected. This was confirmed quantitatively by comparing the axial FWHMs of both methods (Fig. 5c). Our method provided 230 nm axial resolution, which was 2.36 times better than that of the conventional ODT (540 nm). In the experiments with the PMA trimer, normal RBCs, and echinocytes, the enhancement factors of the axial resolution were 1.99, 1.83, and 2.21, respectively (see Fig. S2). The overall results suggest that the axial image resolution can be significantly improved by implementing the proposed method.
Fig. 5 Resolution analysis.
a, b Sliced images of the 3D CSFs obtained from (a) conventional ODT and (b) using our proposed method. Insets, coherent transfer functions used to define CSFs c Axial line profiles of the CSFs along the coloured lines in (a) and (b). FWHM of the conventional ODT and our proposed method are indicated.