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WPT relies on a commercial phase-contrast microscope upgraded with a spatial light modulator (SLM) conjugated to the pupil plane. In our implementation, this hardware is provided by a SLIM module (SLIM Pro, Phi Optics), as shown in Fig. 1a. Figure 1b–d shows the temporal spectrum and autocorrelation properties of the illumination (white-light) field. In addition to the π/2 phase shift between the incident and scattered fields introduced by the objective-phase ring, the SLIM module provides further phase shifts with π/2 increments. At the camera plane, we record three intensity images, corresponding to each phase shift, as illustrated in Fig. 1e, namely
Fig. 1 Optical setup and working principle of Wolf phase tomography (WPT).
a WPT optical setup. The spatial light interference microscopy (SLIM) add-on module is mounted to the output camera port of a phase-contrast microscope. This module shifts the phase between the incident and scattered field with π/2 increments using the SLM. The interference patterns are recorded by the camera. b Spectrum, c autocorrelation, and d second-order derivative of the autocorrelation of the halogen source measured by the spectrometer. e Three phase-shifted frames of 2-μm polystyrene beads are acquired using the SLIM module (×63/1.4 NA)$$ I_d({\bf{r}}) = I_i({\bf{r}}) + I_S({\bf{r}}) + 2\Re [{\Gamma} _{is}\left( {\left\langle \omega \right\rangle \tau _d + {\Delta} \phi ({\bf{r}})} \right)] $$ (1) where $\left\langle \omega \right\rangle \tau _d = - d\pi /2$, d = 1, 2, 3, $\left\langle \omega \right\rangle$ is the central frequency of the incident field, $\Re$ stands for the real part, ${\Delta} \phi$ is the phase difference between the incident field Ui and scattered field US, and ${\Gamma} _{pq}({\bf{r}}_1, {\bf{r}}_2, \tau) = \langle U_p^ \ast ({\bf{r}}_1, t)U_q({\bf{r}}_2, t + \tau)\rangle _t$, $p, q = \{ i, s\}.$ From these three frames, we solve for $\Re \left[ {{\Gamma} _{is}(\left\langle \omega \right\rangle \tau _d + {\Delta} \phi ({\bf{r}}))} \right]$. Based on partially coherent light propagation, governed by the Wolf equations51, the RI of the object can be obtained by (see the full derivation in Supplementary Note 1)
$$ n({\bf{r}}) = \sqrt {\frac{{m({\bf{r}}) - n_0^2\left[ {1 - g({\bf{r}})} \right]}}{{1 + g({\bf{r}})}}} $$ (2) In Eq. (2), the functions m and g are defined as
$$ m({\bf{r}}) = \left. {\frac{{c^2\left( {\nabla ^2\Re [{\Gamma} _{is}({\bf{r}},{\bf{r}},\tau )] + \zeta ({\bf{r}})} \right)}}{{\frac{{\partial ^2\Re [{\Gamma} _{is}({\bf{r}},{\bf{r}},\tau )]}}{{\partial \tau ^2}}}}} \right|_{\tau = - \pi /\left\langle \omega \right\rangle } $$ (3$a$) $$ \zeta ({\bf{r}}) = - 2\Re \mathop {\int}\limits_0^\infty {\langle \nabla U_i^ \ast ({\bf{r}},\omega ) \cdot \nabla U_s({\bf{r}},\omega )\rangle e^{i\omega \pi /\left\langle \omega \right\rangle }d\omega } $$ (3$b$) $$ g({\bf{r}}) = \left. {\frac{{\frac{{\partial ^2\Re [{\Gamma} _{ii}({\bf{r}},{\bf{r}},\tau )]}}{{\partial \tau ^2}}}}{{\frac{{\partial ^2\Re [{\Gamma} _{is}({\bf{r}},{\bf{r}},\tau )]}}{{\partial \tau ^2}}}}} \right|_{\tau = - \pi /\left\langle \omega \right\rangle } $$ (3$c$) where r = (x, y, z) is the spatial coordinate, n0 is the RI of the background media, and c is the speed of light in vacuum. The detailed steps for calculating the terms in Eqs. (3a)–(3c) are given in Supplementary Note 1. The term in Eq. (3b) does not substantially contribute to the final RI and can be omitted for faster construction (see the discussion in Supplementary Note 2). Figure 1b describes the normalized spectrum of a halogen source measured by the spectrometer (ocean optics). The real part of the normalized autocorrelation $\Re [{\Gamma} _{ii}({\bf{r}}, {\bf{r}}, \omega _0\tau)]$ is obtained by taking the Fourier transform of the spectrum (see Fig. 1c). To retrieve the temporal correlation function quantitatively, we normalized the Γii(r, r, 0) value from the spectrometer data to the background intensity from the camera, and corrected it with the spectrally dependent quantum efficiency of the camera. Thus, we ensured that the autocorrelations Γii(r, r, 0) measured by the two different devices have the same value. The second-order time derivative of Γii(r, r, τ) is depicted in Fig. 1d. The Laplacian in Eq. (3a) is calculated using three images with a first-order finite-difference approximation. The z component of the Laplacian was computed using three axially distributed frames separated by a distance that matches the x–y pixel sampling, and is much smaller than the diffraction spot. For example, for a ×40/0.75 NA objective, this distance is 0.14 μm, while the diffraction-limited resolution is 0.4 μm. The second-order derivatives in Eqs. (3a) and (3c) are calculated in MATLAB using three phase-shifted frames. Smaller phase shifts would give more accurate derivatives. However, the contrast between different frames would greatly decrease; thus, the signal-to-noise ratio would decrease as well, resulting in a lower accuracy for the derivative. Therefore, to increase the signal-to-noise ratio and accuracy, we keep the phase increment at π/2. This algorithm requires 40 ms to reconstruct the RI map at one z position with a 3-megapixel field of view (MATLAB, i7-8650U CPU).
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To validate the capability of WPT in extracting the RI distribution, we imaged 2-μm polystyrene microsphere (Polysciences Inc.) z stacks with an RI value of 1.59 at the central wavelength. The beads are suspended in immersion oil (Zeiss) with an RI value of 1.518. Figure 2a shows the three frames corresponding to the different phase shifts of the polystyrene beads. For these experiments, we use a ×63/1.4 NA objective. The real parts of the correlation function Γis at different time-lapse values are illustrated in Fig. 2b. The RI distribution of the microspheres for each z slice is reconstructed via Eq. (2). The 3D rendering of the RI distribution described in Fig. 2c was obtained in AMIRA (Thermo Fisher Scientific). The reconstructed RI value of the microspheres agrees well with the expected value of 1.59 at the central wavelength. The halo artifacts associated with phase-contrast and SLIM images were removed from the final RI maps using our previously reported algorithm52.
Fig. 2 Wolf phase tomography (WPT) on standard samples.
a Three phase-shifted frames of 2-μm polystyrene beads suspended in oil are imaged by spatial light interference microscopy (SLIM) using a ×63/1.4 NA objective. b The real part of the correlation function at three different time delays is obtained by solving Eq. (1). c 3D refractive index (RI) tomogram of the 2-μm polystyrene bead -
A 3D rendering of a bovine sperm cell is displayed in Fig. 3a (Supplementary Video 1). In the sperm head, the acrosome and the nucleus can be identified with RI values between 1.35 and 1.37. The centriole and mitochondria-rich midpiece of the sperm cell yield high RI values (Fig. 3b). The tail of the sperm has an RI value of 1.35, and the axial filament inside the tail, with a slightly higher RI value of 1.36, can be recognized. The end piece of the sperm has the lowest RI value, ~1.34.
Fig. 3 Wolf phase tomography (WPT) of sperm cells.
a 3D refractive index (RI) tomogram of a spermatozoon (×40/0.75 NA objective). b xy-plane projection view. The nucleus, acrosome, centriole, and axial filament of the sperm cell are indicated by white arrows. c xz-plane projection view. d Histogram of the RI of the sperm cell -
Applying the WPT principle, the three frames of hippocampal neurons and their correlation functions are depicted in Fig. 4a and b. The reconstructed RI distribution and 3D rendering of the neurons (Supplementary Video 2) are displayed in Fig. 4c and d. The more detailed structures of individual hippocampal neurons (Supplementary Videos 3 and 4) are illustrated in Fig. 4e and f. The rendering in this case used two colormaps, as shown in Fig. 4e and f. The neuron dendrites have an RI value of ~1.34, while the cell body ranges from 1.35 to 1.38, with a nucleolus of 1.39–1.4. The axon can be recognized in Fig. 4f, as the morphology shows a longer and thinner filamentous structure.
Fig. 4 Wolf phase tomography (WPT) of neurons.
a Three phase-shifted frames of hippocampal neurons (×40/0.75 NA objective). b The real part of the correlation function at three different time lags is solved with Eq. (1). c Refractive index (RI) map of hippocampal neurons. d–f 3D rendering of RI tomograms of hippocampal neurons. e, f Two colormaps are used as indicated to enhance the dendrites and axons. The axon is indicated with a red arrow -
Due to its high throughput, low phototoxicity, absence of photobleaching, and easy sample preparation, WPT is capable of studying real-time volumetric biological events in living cells. We imaged the growth and proliferation of hippocampal neurons over the course of several days in six-well plates typical in phenotypic screening applications. The RI distribution of the whole well of neurons is displayed in Fig. 5a (Supplementary Video 5). One tile zoom-in of the whole well and its distribution of RI is shown in Fig. 5b (Supplementary Video 6). Figure 5c describes the averages of the RI values within this tile versus time. The average RI values increase with time due to neuron growth. Figure 5c illustrates the average RI of the whole tile, including the neurons and background. As the neurons grow, more pixels in the region of interest appear with higher RIs; thus, the average RI becomes larger. Another point worth mentioning is that the range of the y axis in Fig. 5c is from 1.34045 to 1.34070. Thus, due to the averaging over the large field of view, the change in the RI value detected by our system is at the fifth decimal place, indicating the high sensitivity of WPT. Figure 5d shows that the variance of the RI for this tile increases with time as well53. Note that the range of RI variance values is on the order of 10−6, which is detectable due to the sensitivity conferred by the common-path stability and lack of speckles in SLIM.
Fig. 5 Dynamic Wolf phase tomography (WPT) of live cells across multiwell plates.
a The refractive index (RI) map across a whole well of living hippocampal neurons (×10/0.3 NA objective) is composed of 20 × 21 mosaic tiles, each 214 × 204 µm2 in area. b Enlarged RI map of the purple box in (a) with the average (c) and variance (d) of the RI versus time. e Enlarged RI map of the area in the red box in (b) with the average (f) and variance (g) of the RI. The green arrow indicates the increase in the RI when the two neurons separated and their dendrites appeared, and the red arrow shows the decrease in RI when the two neurons died. h Enlarged RI map of the area in the yellow box in (b) with the average (i) and variance (j) of the RI. The green arrow indicates the change in the RI when the dendrites appeared, and the red arrow indicates the decrease in RI when the neuron diedFigure 5e is the enlarged image of the area in the red box in Fig. 5b containing two neurons. The neurons spread out into two regions at ~t = 16 h, continued growing until ~t = 53 h, and then died. We can see that both the average and variance of the RI show three different stages (Fig. 5f, g). One significant change in the average and variance of the RI appeared when the two neurons separated (red arrows). Another change is visible when the two neurons died (green arrows). The death event was accompanied by a decrease in the mean RI, likely due to the membrane permeability, which allowed for water influx.
Figure 5h is a magnified image of the area in the yellow box in Fig. 5b containing one neuron. The neuron dendrites started to appear at ~t = 13 h, resulting in a jump in the average RI (Fig. 5i). The neuron continued growing until ~t = 62 h and then died, leading to a decrease in the average RI. Some oscillations in the variance (Fig. 5j) of the RI appeared before the neuron died and exhibited a clear change after the neuron died. Figure 6 demonstrates the capability of WPT for 3D real-time live-cell imaging. The changes in the morphology of the neuron can be recognized at different time frames.