## Abstract

To probe biological questions with significant biophysical, biochemical, and molecular components, an imaging solution compatible with both endogenous and molecular 3D imaging may be necessary. In this work, we show that structured illumination (SI) microscopy, popularly associated with 3D fluorescent super-resolution, can allow 3D refractive index (RI) reconstructions when operated in the coherent realm. We introduce a novel reinterpretation of coherent SI, which mathematically equates it to a superposition of angled illuminations. Raw acquisitions for standard SI-enhanced quantitative-phase images can be processed into electric field maps of the sample under angled illuminations. Standard diffraction tomography (DT) computation can then be used to reconstruct the sample’s 3D RI distribution at sub-diffraction resolutions. We demonstrate this concept by using SI to computationally reconstruct 3D RI distributions of human breast (MCF-7) and colorectal (HT-29) adenocarcinoma cells. Our experimental setup generates SI patterns using broadband illumination with a spatial light modulator and detects angle-dependent sample diffraction through a common-path, off-axis interferometer with no moving components. This technique may easily pair with SI fluorescence microscopy and important future extensions may include multimodal, sub-diffraction resolution, 3D RI, and fluorescent visualizations.

© 2017 Optical Society of America

## 1. INTRODUCTION

Refractive index (RI) is an important optical property of biological objects that is often exploited to visualize endogenous biological contrast with microscopy techniques that specialize in label-free imaging. Standard examples of such microscopy techniques, which are heavily used in the biological sciences include phase-contrast and differential interference microscopies. Unfortunately, these techniques do not generate distributions that can be quantitatively associated with RI, and thus only allow qualitative biological analysis. More advanced label-free imaging includes the set of quantitative-phase (QP) imaging techniques that generates complex-valued electric field maps encoding optical path length (OPL), the integral of sample RI through the illumination path, into optical amplitude and phase components. However, such QP techniques also do not directly yield biological RI values. Furthermore, though QP distributions are often topographically visualized, they do not offer true 3D visualizations and can be corrupted by out-of-focus sample features [1,2].

In response, several optical techniques have emerged that allow 3D, optically sectioned, biological visualization with quantitative endogenous contrast. Of most relevance to this work is the set of diffraction tomography (DT) techniques that reconstruct 3D RI distributions after illuminating the sample with angled plane waves. Such techniques often rely on either direct holographic detection or, more recently, iterative computational reconstruction [3–5]. Furthermore, due to synthetic aperture principles, this type of DT naturally synthesizes an imaging aperture larger than the physical aperture set by the microscope and thus allows coherent imaging with sub-diffraction resolution [6–10]. Due to its capability for 3D RI visualization, DT has been used to noninvasively probe for single-cell biophysical and biochemical properties, and has been utilized in studies of whole-cell spectroscopy, dry mass, and morphology [9,11,12].

Unfortunately, RI has no inherent molecular specificity, which makes it an unsuitable source of contrast for a host of biological studies that require molecular-specific visualization, such as studies of protein localization, gene expression, organelle dynamics, intercellular transport, etc. [13–15]. For these studies, fluorescence remains the dominant choice for imaging contrast, which allows biological analysis complementary to that allowed by RI imaging. Thus, to study important biological questions that have significant biophysical, biochemical, and molecular components, multimodal RI and fluorescence imaging may be necessary.

Unfortunately, the general DT framework for 3D imaging via angled illuminations is incompatible with fluorescence (due to insensitivity of fluorescent emission to angled excitations). Conversely, confocal, light-sheet, and multiphoton microscopy techniques, which are the current standards for 3D fluorescent imaging, have system designs drastically different from those of DT techniques and are not easily adapted for RI visualization. Thus, conventional fluorescent and DT techniques are not easy candidates for 3D multimodal imaging at single-cell size scales.

We demonstrate here that structured illumination (SI) microscopy, an imaging technique commonly associated with fluorescent super-resolution and 3D fluorescent optical sectioning [16], can also allow 3D reconstruction of RI distributions if operated in the coherent imaging realm. Our technique builds on our previous work on SI-enhanced QP imaging, but reformulates SI’s coherent imaging process into a multiplexed version of tilted plane-wave illumination [17,18]. We start by collecting a sequence of raw acquisitions consisting of sinusoidal sample illuminations with varying spatial frequencies, rotations, and translations, interfered with an off-axis reference wave to extract complex-valued electric field maps. These extracted electric fields under SI illumination were then decomposed into a set of electric fields that would have been measured under single tilted plane-wave illuminations. Standard DT reconstruction was used to combine these computed electric field maps into a 3D RI distribution of the sample. Though we focus on using SI for 3D RI visualization in this work, we emphasize that SI is compatible with both 3D RI and fluorescent imaging, and is thus a promising candidate technique for future 3D multimodal applications.

## 2. FRAMEWORK FOR SI-ENABLED DT

We begin by developing the framework for 3D RI reconstruction within the regime of coherent SI imaging. We start by describing the intuition behind the concept.

#### A. Plane-Wave Decomposition for Coherent SI

In an optical system that images via coherent diffraction, the electric field at the image plane is linearly related to the electric field at the object plane via convolution with the system’s coherent point-spread function (PSF) [19]. Hence, Fourier theory directly shows that the image plane’s electric field frequency spectrum is equivalent to the sample plane’s electric field spectrum after being low-passed by the system’s coherent transfer function (TF). Diffraction theory shows this TF to be a spherical shell, with a circular top-hat projection and a circular arc cross section in lateral and axial frequency space, respectively [20].

When a diffractive object is illuminated by an orthogonal plane wave with a flat phase front, the object plane’s electric field is simply given by the object’s complex transmittance function (CTF). Hence, the 0th spatial frequency (DC) of the object’s diffraction spectrum passes directly through the system’s TF and the image formed will simply be a DC-centered low-passed version of the object’s CTF, as illustrated in Fig. 1(a). For tilted plane-wave illumination, the electric field at the object plane is a product of the object’s CTF with the angled illumination wave front, mathematically described as a phase ramp. From Fourier theory, this results in an angular tilt of the sample’s diffraction into the system’s aperture and a frequency shift (equivalent to the illumination tilt angle) of the object’s diffraction spectrum through the system’s TF.

Thus, the electric field at the image plane will contain spatial frequencies from a DC-offset region of the object’s frequency spectrum, as shown in Fig. 1(b) and 1(c). It follows that SI, which is achieved by interference of tilted coherent plane waves through the object, multiplexes regions of the object’s frequency spectrum through the imaging aperture, with each region corresponding to the angular tilt of an individual wave [Fig. 1(d)]. The tilt angle of these plane waves is directly related to the spatial frequencies present in the SI pattern.

#### B. Coherent SI to Fill Out a 3D Frequency Space

The 3D TF for coherent imaging has been well established by diffraction theory to be a spherical cap mathematically described as a subsection of Ewald’s sphere. The radius and subtended angle are given by the imaging system’s wavenumber and detection numerical aperture (NA). Because this TF has no axial frequency support, a coherent imaging system operating with typical plane-wave illumination (e.g., standard QP microscopes) offers limited axial sectioning capabilities. DT techniques circumvent this issue by sequentially illuminating a 3D object at various angles to shift different regions of the object’s frequency spectrum through the TF. The total frequency content probed by the end of a DT acquisition sequence lies on spherical shells in the object’s frequency spectrum, each of which traces the Ewald’s sphere for each individual illumination wavevector. If a sufficient number of finely incremented illumination angles are used, then the object’s 3D frequency spectrum can be effectively filled, up to the limit allowed by the system’s finite NA. In the case of matched illumination and detection NA, this filled region will form a torus-like shape in 3D frequency space with twice the lateral diffraction limit [8].

Typical DT methods directly scan the illumination angle via a pair of scan mirrors. Following the principles introduced in the previous section, SI can accomplish the same effect by simply imaging sinusoidal patterns onto the object at incremented spatial frequencies. We illustrate this concept in Fig. 2 below. For each spatial frequency, the SI pattern is translated to analytically separate the multiplexed regions of the object’s frequency spectrum, in a manner reminiscent of fluorescent SI. Note that simply illuminating with the highest allowed spatial frequency, as is done in fluorescent SI to maximize resolution gain, is not sufficient to fill out the 3D frequency space in coherent imaging [Fig. 2(b)].

#### C. Extracting Multiplexed Plane-Wave Components

We start here by introducing the mathematical framework, which describes how to extract the multiplexed components from coherent SI (which follows closely from previous work [17,18]). We use off-axis holography to obtain 2D complex-valued electric field measurements from the image plane. As shown in Fig. 3(a), a raw hologram acquired with sinusoidal SI (experimental methods described in the following section) exhibits modulations from both the off-axis interference and SI pattern. The Fourier transform of the interferogram [Fig. 3(b)] shows the central ambiguity term, the electric field spectrum at the image plane, and its conjugate. The image plane spectrum (outlined in yellow) is digitally filtered out and reset to the center of Fourier space [Fig. 3(c)]. The inverse Fourier transform of this gives the electric field at the image plane, which can be mathematically expressed as

In the case of sinusoidal illumination by three-wave interference, we write the illumination field as $i({\mathit{r}}_{\mathit{T}})=1+m\mathrm{cos}({\mathit{k}}_{\mathit{c},\mathit{T}}\xb7{\mathit{r}}_{\mathit{T}}+{\varphi}_{n})$, where $m\le 1$ and $|{\mathit{k}}_{\mathit{c},\mathit{T}}|\le {k}_{c}$. After Fourier-transforming and substituting into Eq. (2), we see that the electric field at the image plane after SI is given as

We see here that the electric field at the image plane is a multiplex of the DC-centered, upshifted, and downshifted object frequency-spectra components, which can be obtained individually with sequential tilted plane waves. This is easily verified by Fourier-transforming the individual 2D expressions for the three component waves in the illumination—${i}_{\mathrm{DC}}({\mathit{r}}_{\mathit{T}})=1,{i}_{+}({\mathit{r}}_{\mathit{T}})=\mathrm{exp}(j{\mathit{k}}_{\mathit{c},\mathit{T}}\xb7{\mathit{r}}_{\mathit{T}})$, and ${i}_{-}({\mathit{r}}_{\mathit{T}})=\mathrm{exp}(-j{\mathit{k}}_{\mathit{c},\mathit{T}}\xb7{\mathit{r}}_{\mathit{T}})$—and substituting into Eq. (2). Yellow arrows in Fig. 3(c) indicate the strong DC signals from these plane waves, which are multiplexed into ${Y}_{\mathrm{SI}}({\mathit{k}}_{\mathit{T}})$. Because ${Y}_{\mathrm{SI}}({\mathit{k}}_{\mathit{T}})$ is a linear summation of the object’s frequency-shifted terms, three translations of the SI pattern, i.e., $\{{\varphi}_{n}|n=\mathrm{0,1},2\}$, allows a linear solution of the raw component terms $H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}),H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}+{\mathit{k}}_{\mathit{c},\mathit{T}})$, and $H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}-{\mathit{k}}_{\mathit{c},\mathit{T}})$. Here $H({\mathit{k}}_{T})$ acts as a DC-centered window over the shifted regions of the object’s spectrum. Correcting for these shifts results in the terms $H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}),H({\mathit{k}}_{\mathit{T}}-{\mathit{k}}_{\mathit{c},\mathit{T}})X({\mathit{k}}_{\mathit{T}})$, and $H({\mathit{k}}_{\mathit{T}}+{\mathit{k}}_{\mathit{c},\mathit{T}})X({\mathit{k}}_{\mathit{T}})$, which effectively summarize the objective of SI to probe different regions of an object’s spectrum with diffraction-limited windows. Inverse Fourier-transforming these corrected terms yields the background-subtracted electric field QP maps ${x}_{\mathrm{DC}}({\mathit{r}}_{\mathit{T}}),{x}_{+}({\mathit{r}}_{\mathit{T}})$, and ${x}_{-}({\mathit{r}}_{\mathit{T}})$. We show both the raw component terms and the associated QP maps in Fig. 3(d)–3(f). Note that ${x}_{\mathrm{DC}}({\mathit{r}}_{\mathit{T}})$ is exactly the output expected from standard QP microscopy with orthogonal illumination.

#### D. Tomographic Reconstruction of 3D RI

The mathematical framework discussed in this section largely adapts the DT framework, originally introduced in [21,22], to reconstruct 3D RI distributions from 2D measurements taken with coherent SI. We strive to keep similar notation with [3] specifically to allow easy comparison. We first introduce the scattering potential, which gives information about the object’s RI:

where $\mathit{r}=(x,y,z)$ is the 3D spatial vector, ${k}_{\lambda}=2\pi /\lambda $ is the free-space wavenumber, $n(\mathit{r})$ is the object’s 3D RI distribution, and ${n}_{m}$ is the constant media RI. The scattering potential is formulated directly from the wave equation under the implicit assumption that the object is semi-transparent, such that the total diffracted wave after passing through the sample, $u(\mathit{r})$, is a sum of the incident, ${u}_{i}(\mathit{r})$, and sample-specific scatter, ${u}_{s}(\mathit{r})$, waves, respectively:Approximating the object as a lightly scattering sample, where ${u}_{i}(\mathit{r})\gg {u}_{s}(\mathit{r})$, we can express the sample-specific scatter wave as

As shown in the previous section, SI enables decoupling of individual, background-subtracted, electric field maps—${x}_{\mathrm{DC}}({\mathit{r}}_{\mathit{T}})$, ${x}_{+}({\mathit{r}}_{\mathit{T}})$, and ${x}_{-}({\mathit{r}}_{\mathit{T}})$. These maps could be equivalently obtained by sequential illumination with the plane-wave components forming the SI, say, ${i}_{\mathrm{DC}}(\mathit{r})=\mathrm{exp}(j{\mathit{k}}_{\mathbf{DC}}\xb7\mathit{r})$, ${i}_{+}(\mathit{r})=\mathrm{exp}(j{\mathit{k}}_{+}\xb7\mathit{r})$, and ${i}_{-}(\mathit{r})=\mathrm{exp}(-j{\mathit{k}}_{-}\xb7\mathit{r})$. In the example presented in the previous section, where a SI pattern with a 2D spatial-frequency vector ${\mathit{k}}_{\mathit{c},\mathit{T}}$ was used, the wavevectors for these plane waves are ${\mathit{k}}_{\mathrm{DC}}=(\mathrm{0,0},{k}_{\lambda})$, ${\mathit{k}}_{+}=({\mathit{k}}_{\mathit{c},\mathit{T}},{k}_{c,z})$, and ${\mathit{k}}_{-}=(-{\mathit{k}}_{\mathit{c},\mathit{T}},{k}_{c,z})$, where ${k}_{z}=\sqrt{{({n}_{m}{k}_{\lambda})}^{2}-{|{\mathit{k}}_{\mathit{c},\mathit{T}}|}^{2}}$. Thus, each background-subtracted electric field reconstruction, which we now generally refer to as ${x}_{m}({\mathit{r}}_{\mathit{T}})$, can be associated with an effective illumination wavevector, say, ${\mathit{k}}_{m}$. For a complete SI data set with acquisitions taken with sinusoidal SI patterns undergoing $s$ rotations with $t$ increments in spatial frequency per rotation, we build a collection of electric field maps with associated illumination wavevectors—$\{{x}_{m}({\mathit{r}}_{\mathit{T}}),{\mathit{k}}_{m}|m=1,2,\dots 3\xb7s\xb7t\}$.

Under the Rytov approximation, ${U}_{s}({\mathit{k}}_{\mathit{T}}+{\mathit{k}}_{0,\mathit{T}};z=0)=\mathcal{F}\{\mathrm{log}({x}_{m}({\mathit{r}}_{\mathit{T}}))\}$ for each illumination wavevector ${\mathit{k}}_{0}={\mathit{k}}_{m}$, where $\mathcal{F}\{\xb7\}$ and $\mathrm{log}(\xb7)$ are the Fourier and complex logarithm operators, respectively [3]. Thus, for each $1\le m\le 3\xb7s\xb7t$, the values of $F(\mathit{k})$ along spherical surfaces in 3D frequency space can be reconstructed by substituting $\mathcal{F}\{\mathrm{log}({x}_{m}({\mathit{r}}_{\mathit{T}}))\}$ and ${\mathit{k}}_{m}$ for ${U}_{s}({\mathit{k}}_{\mathit{T}}+{\mathit{k}}_{0,\mathit{T}};z=0)$ and ${\mathit{k}}_{0}$, respectively, in Eq. (8) above. We illustrate this process in Fig. 4. After $F(\mathit{k})$ is sufficiently reconstructed, an inverse Fourier transform directly gives back the estimate for $f(\mathit{r})$. The 3D RI can then be solved easily using Eq. (4).

## 3. EXPERIMENTAL METHODS AND RESULTS

We experimentally demonstrated SI’s DT capabilities by using a variant of our original SI-DPM system introduced in [18]. Our current optical system, illustrated in Fig. 5 below, used broadband, single-mode illumination at $\lambda =488\pm 15\text{\hspace{0.17em}}\mathrm{nm}$ (NKT Photonics, EXW-6). The SI pattern was generated by programming a sinusoidal pattern onto an amplitude spatial light modulator (SLM, Holeye, HED 6001), which was then imaged through a polarization beam splitter (PBS, Thorlabs, PBS 251) onto the object through a system of lenses. An adjustable iris diaphragm ($F$) placed at a Fourier conjugate plane to the SLM spatially filtered out all diffraction orders except the $\pm 1$ and 0 orders. These three orders were focused through the condenser objective lens (OBJ, 63X, 1.4 NA, Zeiss) to create a three-beam sinusoidal interference onto the sample. The diffraction from the sample was imaged in transmission through another OBJ (63X, 1.4 NA, Zeiss) into a conventional diffraction-phase setup. A Ronchi grating (RG, Edmund Optics, Ronchi 70 lpmm) split the signal into three main diffraction orders. A mask, which included a 20 μm pinhole (PH, Edmund Optics, 52-869), was used to spatially filter the 0th order to create a common-path, phase-stable, off-axis reference beam to interfere with the 1st order and create an off-axis hologram at the camera plane. The $-1$st order (and all other extraneous orders arising from the RG) was blocked by the mask. We note that, in the case of general widefield imaging (when all pixels on the SLM are turned on), the 0th and $\pm 1$st orders arising from the RG each contained only one spatial-frequency component. In the case of SI, however, the 0th and $\pm 1$st orders from the RG each individually contained the spatial-frequency components present in the SI pattern, i.e., the 0th and $\pm 1$st orders from the sinusoidal pattern written onto the SLM. Due to the spectrally broadband nature of the illumination, each spatial-frequency component in the plane of the PH was spectrally dispersed with respect to its order number in relation to the RG and SLM. However, the central 0th SLM/RG-order component represents the purely transmitted portion of the illumination, which remained undispersed and physically static regardless of the rotation or spatial frequency magnitude of the SI pattern written onto the SLM. Thus, this component was suitable for spatial filtering by PH [Fig. 5(c)].

Our diffraction-limited lateral coherent resolution (defined as the minimum resolvable spatial period) for typical QP imaging is expected to be $\lambda /\mathrm{NA}\approx 350\text{\hspace{0.17em}}\mathrm{nm}$. Due to identical condenser and detection lenses, the SI patterns are also diffraction-limited. Thus, the final RI reconstructions are expected to have lateral resolution double that of typical QP, and to be able to resolve spatial frequencies of period $\lambda /2\mathrm{NA}\approx 180\text{\hspace{0.17em}}\mathrm{nm}$. The resolvable period of axial spatial frequencies is expected to be $\lambda /n(1-\mathrm{cos}\text{\hspace{0.17em}}\theta )\approx 520\text{\hspace{0.17em}}\mathrm{nm}$, where $n=1.51$ and $\theta $ are, respectively, the immersion index and maximum accepted half-angle of light, such that $\mathrm{NA}=n\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta $. To generate a 3D RI distribution, 960 raw holograms were acquired (SI patterns underwent 5 translations per spatial frequency, 32 spatial frequencies per rotation, and 6 rotations; see Visualization 1) with 15 ms per acquisition. Periodic reconstruction artifacts associated with SI reconstruction were notch-filtered out in frequency space [26]. Here we note that from a theoretical standpoint, Eq. (3) demonstrates that only three translations of a SI pattern, per spatial frequency, per rotation, were necessary to reconstruct $H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}),H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}+{\mathit{k}}_{\mathit{c},\mathit{T}})$, and $H({\mathit{k}}_{T})X({\mathit{k}}_{\mathit{T}}-{\mathit{k}}_{\mathit{c},\mathit{T}})$. We, however, used five translations of each SI pattern to further decrease background noise. Experimentally, RI fluctuations due to background noise were measured to have a standard deviation of $\sigma =2.20\times {10}^{-4}$.

#### A. QP versus RI Visualization of Polystyrene Microspheres

To verify SI’s capabilities to quantitatively reconstruct 3D RI distributions, we imaged a sample consisting of a monolayer of 7.3 μm polystyrene microspheres immersed in oil [with calibrated ${n}_{m}(\lambda )=1.594$ at $\lambda =488\text{\hspace{0.17em}}\mathrm{nm}$]. Figures 6(a) and 6(b) show the in-focus lateral visualization of a select sample region after QP and RI reconstruction, respectively. Figures 6(c) and 6(d) compare the 3D optical sectioning capabilities between the two reconstruction schemes by taking an axial cross section from the region indicated by the dashed white line in Figs. 6(a) and 6(b), respectively [the 3D QP signal was reconstructed by digitally propagating the in-focus signal in Fig. 6(a) using Fresnel kernels]. As is evident, RI reconstruction demonstrates clear optical sectioning by depth-localizing the RI signal from the microsphere. Conversely, the QP signal shows no such depth localization. We note that, though Fig. 6(b) demonstrates a clear circular RI cross section laterally, Fig. 6(d) shows that the axial RI cross-cut is elongated axially. This deformation is due to the “missing-cone” problem, which results from limited axial frequency-space coverage for the lower lateral frequencies. This issue is inherent to beam-scanning DT techniques and is typically addressed with computational methods [27]. In this work, we simply use an iterative non-negativity constraint to computationally fill this missing information [3]. Figure 6(e) and 6(f) show the filling (before addition of the non-negativity constraint) of the sample’s scattering potential from lateral and axial viewpoints, respectively, in 3D frequency space. 3D visualization [Fig. 6(g)] directly shows the torus shape commonly associated with DT. From this, illumination angles incident on the sample were measured to range from $-65\xb0$ to $+65\xb0$. This directly resulted in experimental lateral and axial resolutions of $\sim 190\text{\hspace{0.17em}}\mathrm{nm}$ and $\sim 545\text{\hspace{0.17em}}\mathrm{nm}$, respectively.

We now quantitatively evaluate the reconstructed QP and RI values. Figure 6(h) shows the QP profile measured across the microsphere indicated by the dashed white line in Fig. 6(a); it shows a QP peak at $\sim 0.6\text{\hspace{0.17em}}\mathrm{rad}$. This profile matches the theoretical QP profile expected from a microsphere with a RI of ${n}_{\mathrm{PS}}=1.601$. This measured RI value falls within 1% of the theoretical RI value for polystyrene, $\widehat{{n}_{PS}}(\lambda )=1.605$ for $\lambda =488\text{\hspace{0.17em}}\mathrm{nm}$, based on previous work fitting spectral RI measurements with the Sellmeier dispersion formula [28].

Unlike the QP signal, which integrated RI along the OPL and demonstrated an elliptical profile in Fig. 6(h) for a spherical particle with homogenous RI, reconstructed RI values are localized to specific points in 3D space. Thus, the RI profile across a cross section of a microsphere is expected to take the form of a rectangular function bound between the RI of the polystyrene microsphere and the surrounding media. Figure 6(i) shows the measured RI profile across the same region measured in Fig. 6(h)and clearly demonstrates a rectangular profile bound between ${n}_{\mathrm{PS}}$ and ${n}_{m}$.

#### B. Biological Imaging

### 1. 3D RI Reconstruction of Human Breast Adenocarcinoma

We first demonstrate 3D biological RI reconstruction of cells from the human breast adenocarcinoma (MCF-7) line. These cells are popularly used in studies of tumor biology and previous works have extensively studied their signaling pathways, gene activation events, molecular receptors, hormonal responsiveness, and proliferation rates [29]. MCF-7 cells were cultured in Dulbecco’s modified Eagle’s medium (DMEM) supplemented with 10% fetal bovine serum, and 10 μg/ml and 1 μl/ml pen-strep. Cells were plated at low density onto #1.5 coverslips and allowed to attach overnight. The cells were subsequently fixed using 4% formaldehyde in phosphate-buffered saline for 10 min and then washed and incubated with phosphate-buffered saline prior to imaging.

In Fig. 7, we compare 3D RI reconstruction with typical QP visualization of a single MCF-7 cell at axial slices spaced 2.26 μm through the cell volume. Standard Fresnel propagation kernels were used to digitally propagate the 2D QP map. We see immediately from both RI [Fig. 7(a)–7(d)] and QP [Fig. 7(e)–7(h)] visualizations that the specific imaged cell has a high-mass bulk center region with low-mass surrounding cytoplasmic extensions. However, as shown in Fig. 7(e)–7(h), QP does not allow high-contrast visualization of structures within the high-mass region, and, in fact, does not allow the viewer to even make an educated guess about whether the region is simply “thick,” which relates to cell morphology, or “dense,” which relates to cell composition. This is of course due to RI and physical path length being entangled into OPL, the source of contrast in QP imaging. Furthermore, the various axial slices show almost no variation in QP signal, as shown in Fig. 7(e), which indicates two cytoplasmic extensions, which are visualized even after axial defocus. This conforms to our expectation that QP imaging has little axial resolution. In contrast, RI visualizations of the same axial slices show a drastic increase in intracellular contrast, which can be attributed to both optical sectioning and the doubled lateral resolution over QP imaging. An obvious advantage of RI imaging includes significant enhancement of image contrast when visualizing the central high-mass region of the cell. Unlike the ambiguities associated with QP imaging, RI shows that this region has a higher “density” (corresponding to RI values of $\sim 1.335$) than the surrounding cytoplasm, which suggests a compositional change. In fact, Fig. 7(c) shows that this region has a circular delineation and is markedly denser than the surrounding cytoplasm, which suggests that this region is dominated by one large subcellular component. After referring to previous works, which show molecular-specific fluorescent labeling of MCF-7 cells, and drawing comparisons to the size, shape, and positioning of the high-density region, it is reasonable to hypothesize that this high-density region is the cell’s nucleus. Even within the nucleus, high-density structures can be clearly visualized. Given the shapes and locations of these structures, we hypothesize that they are the nucleolus and endoplasmic reticulum. The nucleolus shows a grainy texture in Fig. 7(c), which conforms to previous works visualizing MCF-7 nucleoli via fluorescence [30]. The surrounding cytoplasm also exhibits significantly increased contrast and various cytoplasmic extensions and cavities are clearly visible in sharp focus in Fig. 7(a)–7(c). These cytoplasm structures were either not visible or out of focus with QP imaging [Fig. 7(e)–7(h)]. Furthermore, RI visualization shows both the nucleus and cytoplasm changing shape as the imaged depth plane is axially moved, which demonstrates 3D optical sectioning (see Visualization 2).

Interestingly, we have observed (not shown here) MCF-7 cells from the same population, which exhibit nuclear RI values lower than those of the rest of the cell. Furthermore, most MCF-7 cells do not have as high a karyoplasmic ratio (nucleus–whole-cell volume ratio). Indeed, one of the fundamental properties of eukaryotic cells is an ability to maintain nuclear sizes in relation to whole-cell volume, which is crucial for many nuclear-specific cell functions, such as nuclear transport, gene expression, organization of inter-nuclear components and compartments (nucleolus, Cajal bodies, Kremer bodies, etc.), and general cell regulation [31]. Given the expected consistency of nuclear size, nuclear growth is typically associated with a progression through the cell cycle [32]. We hypothesize that the enlarged nucleus in Fig. 7 indicates this individual cell’s preparation, before fixation, to enter into cell division.

### 2. 3D RI Reconstruction of Human Colorectal Adenocarcinoma

We now demonstrate 3D biological RI reconstruction of human colorectal adenocarcinoma (HT-29) cells. Like the MCF-7 cells presented in the previous section, the HT-29 cell line is widely used to study tumor biology. However, HT-29 cells have a drastically different morphology than MCF-7 and typically exhibit globular shapes. HT-29 cells have been popular imaging choices in previous DT works [3,33], perhaps because of their rounded, roughly spherical profiles, and so we also image them to further explore 3D RI reconstruction via SI. HT-29 cells were grown in McCoy’s 5A Modified Medium supplemented with 10% fetal bovine serum and 1 μl/ml pen-strep antibiotic. Subsequent fixation steps with formaldehyde were performed in a similar fashion as to the MCF-7 cells above.

In Fig. 8, we demonstrate 3D RI reconstruction of a pair of conjoined HT-29 cells and visualize their hallmark globular structure (see Visualization 3 for a comparison between RI and QP 3D reconstruction). In Fig. 8(a)–8(d), we show axial slices taken through the RI tomogram taken at 3.8 μm increments. Due to the cells’ globe-like profiles, clear differences in the axial slices are apparent. The cell boundary separating the cell cytoplasm from the surrounding phosphate-buffered saline immersion media is clearly visible in all the slices and shows the cells’ circular cross sections enlarging and then shrinking as the visualized cross section shifts axially up through the cell, as is expected when visualizing a globular structure. Figure 8(a) shows an axial slice at $z=-5.67\text{\hspace{0.17em}}\mathrm{\mu m}$, which contains a number of high-density localizations below the nucleus. Molecular labeling is necessary to truly identify these localizations. However, given their sizes and positions around the nucleus, we hypothesize that these localizations are lipid-based vesicles, which are known to have a high-RI lipid bilayer. Previous works have demonstrated fluorescent imaging of vesicles and have reported them to congregate adjacent to the nuclear region of the cell [34]. These findings conform to our observation of the vesicles being situated almost directly beneath the nucleus. The next axial slice at $z=-1.89\text{\hspace{0.17em}}\mathrm{\mu m}$ is closer to the center of the cell and exhibits a larger HT-29 cross-sectional area. Here, we can see a ring-shaped region of high RI, which directly surrounds the nuclear region.

We also see that the tail ends of the nucleoli start to appear at this axial position. Interestingly, unlike our observations with MCF-7 cells shown in Fig. 7, even though the nuclei in the two HT-29 cells show regions with modestly high RIs, the nuclei as whole organelles do not exhibit a distinct difference in RI compared with the rest of the cell.

Previous studies performing 3D RI tomography on HT-29 cells have demonstrated similar observations [3]. This observation is continued into the next axial slice, at $z=+1.89\text{\hspace{0.17em}}\mathrm{\mu m}$, where now the nucleoli appear fully formed. In Fig. 8(e)–8(g), we show 3D visualization of the RI tomogram after applying RI-thresholded opacity constraints so as to emphasize different biological features. Figure 8(e) emphasizes the cell boundary, which encapsulates the cell. The cells’ subcellular components can be viewed through the boundary so as to visualize their positioning in the cell body. Figure 8(f) emphasizes the intracellular content within the cytoplasm by setting the RI values associated with the cytoplasm ($\mathrm{RI}<1.337$) transparent. We can see that intracellular components with moderately high RIs, typically $\sim 1.35$, are concentrated in areas around the nuclear region. Figure 8(g) shows the positioning of the high-RI components ($\mathrm{RI}>1.37$) to be localized to either the nucleoli or nuclei periphery. A rotational view of the tomogram is shown in Visualization 4.

## 4. DISCUSSION AND CONCLUSION

In this work, we have presented a mathematical framework, which draws parallels between coherent SI and DT by first portraying SI as a multiplexed version of tilted plane-wave illumination. We emphasize that the electric field measured at the image plane after SI of the object is a superposition of the electric fields that would have been measured after illuminating with the tilted plane waves individually composing the SI pattern. It follows that SI of the object at various rotations and spatial frequencies is sufficient to fill out a 3D frequency space, as is required in conventional DT. However, unlike conventional DT techniques that utilize scanning mirrors (associated with mechanical instabilities and illumination-angle ambiguities) and/or separate off-axis reference arms (associated with temporal phase instabilities), SI enables phase-stable electric field acquisitions with a common-path, off-axis setup with no mechanically moving components. Furthermore, the common-path, off-axis interference configuration also allows spectrally broadband illumination to be used, which improves imaging artifacts from coherent noise. This resistance to temporal instabilities and coherent noise distinguishes our work from a very recent report utilizing time-multiplexed averaging of structured patterns for RI reconstruction [36]. Though the procedure followed in the report showed similarities to the framework presented here, the report’s experimental implementation required a coherent, monochromatic laser imaging through a conventional Mach–Zehnder setup, as well as temporal averaging to minimize digitation artifacts from the system’s digital micromirror device (DMD). Thus, such an implementation may be subject to temporal instabilities and coherent noise, as well as lower RI acquisition rates.

This work presents SI as a potential solution for 3D RI measurements with high temporal stability and low coherent noise. Though the optical system presented in this work generated SI patterns using an SLM based on nematic liquid crystal technology (which associates with slow response times), we fully expect future improvements of this system to incorporate ferroelectric-based SLMs or DMDs for significantly faster generation of SI patterns.

Furthermore, we also note that our theoretical treatment for SI-enabled DT assumed monochromatic illumination. In reality, however, we used illumination with $\sim 30\text{\hspace{0.17em}}\mathrm{nm}$ bandwidth, which was previously demonstrated to reduce coherent noise in 2D QP imaging [18]. More notably for 3D RI imaging, broadband illumination also results in imaging a 3D frequency space through an Ewald shell of non-uniform, non-infinitesimal axial thickness [33]. It follows that simply broadening the illumination spectrum may allow improvements in axial resolution that could potentially be utilized in conjunction with tilted illuminations [37]. Future extensions of this work may explore optimization of illumination bandwidth with number of illumination tilt angles to efficiently fill the 3D frequency space.

Generally, this work demonstrates that SI, which is popularly associated with fluorescent super-resolution imaging, also has significant utility for coherent biological imaging applications. Indeed, coherent imaging is uniquely capable of imaging biological objects with high endogenous contrast and has been used to extract biophysical and biochemical properties. Applications of this have included quantitative measurements of whole-cell morphology and mass, spectroscopy, hemoglobin concentration, etc. [9,11,12]. Previous work, which considered SI’s capabilities to augment the effectiveness and accuracy of these applications, utilized SI for sub-diffraction resolution imaging of coherent scatter and optical phase delays [17,18,38,39]. To the best of our knowledge, this work reports one of the earliest examples of applying the SI framework for extracting 3D biological RI [36].

Furthermore, this method to extract 3D biological RI lends itself easily to compatibility with SI-based fluorescent super-resolution. Fluorescence imaging is the dominant choice for molecular-specific biological visualization and is the standard imaging technique used in a host of works studying gene expression, protein interaction, organelle structure, subcellular transport, and general intracellular dynamics [13–15]. As such, fluorescent imaging offers imaging capabilities that directly complement those offered by coherent imaging, which offer no intrinsic specificity. Fluorescent SI has demonstrated that 3D fluorescent resolution doubling further augments such imaging capabilities and has been used for visualization of centrosome and nuclear architecture, as well as microtubule and mitochondrial dynamics [16,40].

Based on this work, we believe that SI holds promise as a multimodal sub-diffraction technique that can be used for 3D high-resolution fluorescent and RI volume imaging. Such a technique can have important applications in studying biological questions that have significant molecular and biophysical/biochemical components.

## Funding

National Science Foundation (NSF) (1403905).

## Acknowledgment

We thank Kyoohyun Kim for useful discussions about diffraction tomography and Brianna Loomis for help in preparation of biological samples.

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