We demonstrate a label-free, scan-free intensity diffraction tomography technique utilizing annular illumination (aIDT) to rapidly characterize large-volume three-dimensional (3-D) refractive index distributions |

## 1.## IntroductionThree-dimensional (3-D) refractive index (RI) distributions of cells and tissues are useful for the morphological detection and diagnosis of disease in biomedical imaging. Due to the low absorption and contrast of biological samples in the visible spectrum, exogenous labels (e.g., fluorophores) are commonly used as biomarkers to visualize regions of interest. For example, confocal fluorescence and two-photon microscopy are commonly used when imaging thick 3-D samples. Although these methods provide excellent optical sectioning, the excitation light and contrast agents for fluorescence imaging can cause photobleaching, phototoxicity, and other damaging effects that artificially alter the sample’s behavior and structure. The most widely used interferometry-based RI tomography technique is optical diffraction tomography (ODT). In ODT, the scattered field is directly measured from digitally recorded interferograms taken under different illumination angles (i.e., phase projection measurement). Several ODT approaches have been applied in biomedical studies for evaluating cell physiology, The alternative approach we demonstrate here uses intensity-only measurements for QPI based on the principle of IDT. To improve IDT’s temporal resolution, we develop a fast and accurate annular illumination IDT (aIDT) technique overcoming these limitations. Importantly, our technique reduces the data requirement by more than $60$ times, achieving more than 10-Hz for imaging a volume of $\sim 350\text{\hspace{0.17em}}\mu \mathrm{m}\times 100\text{\hspace{0.17em}}\mu \mathrm{m}\times 20\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, with near diffraction-limited lateral resolution of 487 nm and axial resolution of $3.4\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ in the 3-D RI reconstruction. These improvements enable Several hardware and algorithmic innovations enable our annular IDT technique. First, our hardware employs a programmable LED ring consisting of only 24 surface-mounted LEDs. Compared to existing LED matrix-based systems, ## 2.## Methods## 2.1.## aIDT PrincipleOur aIDT imaging system combines an LED ring illumination unit and a standard bright-field microscope [Fig. 1(a)]. The LED ring is placed some distance $h$ away under the sample, as illustrated in Fig. 1(b). Importantly, the distance is carefully tuned such that the ring is matched with the perimeter of the objective lens’s pupil aperture. This can be done because this illumination design approximately follows the Köhler geometry, in which each LED provides a plane wave of unique angle. Denoting the radius of the ring as $r$, the illumination NA (${\mathrm{NA}}_{\mathrm{illum}}$) of each LED is set by ${\mathrm{NA}}_{\mathrm{illum}}=r/\sqrt{{r}^{2}+{h}^{2}}$. Hence, one can achieve ${\mathrm{NA}}_{\mathrm{illum}}$ matching the objective NA (i.e., ${\mathrm{NA}}_{\mathrm{obj}}={\mathrm{NA}}_{\mathrm{illum}}$) by simply adjusting the LED height $h$. During the experiment, we acquire up to 24 images capturing bright-field intensity measurements from each individual LED in our ring unit [Fig. 1(c)]. By downsampling the total LED number, we can improve our acquisition speed to accommodate dynamic live samples. The intensity of each image results from the interference of the scattered field from a weakly scattering object and the unperturbed illuminating field [Fig. 1(c)]. By quantifying the Fourier space information under the first Born approximation, we derive phase and absorption transfer functions (TFs) Several important observations govern our illumination design. First, each TF’s Fourier coverage features a pair of circular regions describing the scattered field’s information and its complex conjugate. The system’s objective ${\mathrm{NA}}_{\mathrm{obj}}$ defines the circles’ radius while the illumination angle ${\mathrm{NA}}_{\mathrm{illum}}$ defines the center positions. By matching ${\mathrm{NA}}_{\mathrm{illum}}$ to ${\mathrm{NA}}_{\mathrm{obj}}$, one ensures maximizing the Fourier coverage allowed by the system (i.e., $2{\mathrm{NA}}_{\mathrm{obj}}$), as demonstrated in Fig. S1 in the
Supplementary Material. Second, the phase information is captured by the antisymmetric Fourier information whereas the absorption information is by symmetric information [Fig. 1(d)]. The antisymmetry in the phase TF further indicates that any overlap between the two circular regions (by using ${\mathrm{NA}}_{\mathrm{illum}}<{\mathrm{NA}}_{\mathrm{obj}}$) will cancel the captured low-spatial frequency information [Fig. S1(a) in the
Supplementary Material]. ## 2.2.## Transfer Function AnalysisIn the aIDT forward model, a 3-D sample is discretized into a stack of 2-D sample slices. Following the IDT derivation, ## Eq. (1a)$${H}_{P}(\mathbf{u})=\frac{i\mathrm{\Delta}z{k}_{0}^{2}}{2}(P(\mathbf{u}-{\mathbf{\rho}}_{\mathbf{s}})\frac{\mathrm{exp}\{-i[\eta (\mathbf{u}-{\mathbf{\rho}}_{\mathbf{s}})-{\eta}_{\mathbf{s}}]z\}}{\eta (\mathbf{u}-{\mathbf{\rho}}_{\mathbf{s}})}-P(\mathbf{u}+{\mathbf{\rho}}_{\mathbf{s}})\frac{\mathrm{exp}\{i[\eta (\mathbf{u}+{\mathbf{\rho}}_{\mathbf{s}})-{\eta}_{\mathbf{s}}]z\}}{\eta (\mathbf{u}+{\mathbf{\rho}}_{\mathbf{s}})}),$$## Eq. (1b)$${H}_{A}(\mathbf{u})=-\frac{\mathrm{\Delta}z{k}_{0}^{2}}{2}(P(\mathbf{u}-{\mathbf{\rho}}_{\mathbf{s}})\frac{\mathrm{exp}\{-i[\eta (\mathbf{u}-{\mathbf{\rho}}_{\mathbf{s}})-{\eta}_{\mathbf{s}}]z\}}{\eta (\mathbf{u}-{\mathbf{\rho}}_{\mathbf{s}})}+P(\mathbf{u}+{\mathbf{\rho}}_{\mathbf{s}})\frac{\mathrm{exp}\{i[\eta (\mathbf{u}+{\mathbf{\rho}}_{\mathbf{s}})-{\eta}_{\mathbf{s}}]z\}}{\eta (\mathbf{u}+{\mathbf{\rho}}_{\mathbf{s}})}),$$P is the objective pupil function, ${\mathbf{\rho}}_{\mathbf{s}}$ is the lateral illumination wave vector, $\eta (\mathbf{u})=\sqrt{{k}_{0}^{2}-{|\mathbf{u}|}^{2}}$ is the axial wave vector, ${\eta}_{\mathbf{s}}({\mathbf{\rho}}_{\mathbf{s}})=\sqrt{{k}_{0}^{2}-{|{\mathbf{\rho}}_{\mathbf{s}}|}^{2}}$ is the illumination axial wave vector, and z is the sample’s axial location. With slice-wise discretization, we denote the axial location as $\mathit{z}=\mathit{m}\mathrm{\Delta}z$, where m is the slice index for M slices.## 2.3.## Three-Dimensional RI Reconstruction AlgorithmTo achieve 3-D RI reconstruction, aIDT solves an inverse problem through deconvolution. First, each intensity image is processed to remove the background. Next, the Tikhonov regularized deconvolution is performed to reconstruct the object’s real and imaginary RI. The main idea of our slicewise deconvolution process is to replace the continuous integration along the axial direction by a discrete sum over the slice index. Importantly, since the TFs are independent between slices, the scattering information from different sample slices is also decoupled. This decoupling allows us to use a computationally efficient, slice-based deconvolution procedure to reconstruct the cross-sectional RI distribution one slice at a time. The slice spacing is chosen arbitrarily during the computation. The achieved axial resolution is characterized by analyzing the reconstructed stack and is found to approximately match with the diffraction limit, set by $\lambda /({n}_{m}-\sqrt{{n}_{m}^{2}-{\mathrm{NA}}_{\mathrm{obj}}^{2}})$, where ${n}_{m}$ is the RI of surrounding medium. The closed-form solutions for real part of the permittivity contrast (i.e., phase) $\mathrm{\Delta}{\epsilon}_{\mathrm{Re}}$ and imaginary part of the permittivity contrast (i.e., absorption) $\mathrm{\Delta}{\epsilon}_{\mathrm{Im}}$ at each axial slice are ## Eq. (2a)$$\mathrm{\Delta}{\epsilon}_{\mathrm{Re}}[m]={\mathit{F}}^{-1}\left(\frac{1}{A}\right\{[\sum _{l}{|{H}_{A}(l,m)|}^{2}+\beta ]\odot [\sum _{l}{H}_{P}^{*}(l,m)\odot \tilde{g}[l]]-[\sum _{l}{H}_{P}^{*}(l,m)\odot {H}_{A}(l,m)]\odot [\sum _{l}{H}_{A}^{*}(l,m)\odot \tilde{g}[l]]\left\}\right),$$## Eq. (2b)$$\mathrm{\Delta}{\epsilon}_{\mathrm{Im}}[m]={\mathit{F}}^{-1}\left(\frac{1}{A}\right\{[\sum _{l}{|{H}_{P}(l,m)|}^{2}+\alpha ]\odot [\sum _{l}{H}_{A}^{*}(l,m)\odot \tilde{g}[l]]-[\sum _{l}{H}_{P}(l,m)\odot {H}_{A}^{*}(l,m)]\odot [\sum _{l}{H}_{P}^{*}(l,m)\odot \tilde{g}[l]]\left\}\right),$$## 2.4.## System SetupOur aIDT setup is built on an upright bright field microscope (E200, Nikon) and replaces the existing illumination unit with a ring LED (1586, Adafruit). The radius of the ring LED unit is $\sim 30\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$. The LED ring is placed $\sim 35\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$ away from the sample, whose center is aligned with the optical axis of the microscope. Each LED approximately provides spatially coherent quasimonochromatic illumination with central wavelength $\lambda =515\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ and $\sim 20\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ bandwidth. All experiments in the main text were conducted using a $40\times $ microscope objective (MO) (0.65 NA, CFI Plan Achro). Images were taken with an sCMOS camera (PCO. Panda 4.2, $6.5\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ pixel size), which is synchronized with the LED source to provide camera-limited acquisition speed. The LED ring unit is driven by a microcontroller (Arduino Uno). In addition, we provide additional results using a $10\times $ (0.25 NA, CFI Plan Achro) MO to image spiral algae (S68786, Fisher Scientific) in Video S1, MPEG, 7.1 MB [URL: https://doi.org/10.1117/1.AP.1.6.066004.7]. During the experiments imaging living ## 2.5.## Self-Calibration MethodDuring the reconstruction, we first perform a numerical self-calibration procedure. The assumed illumination angles for the ring LED do not necessarily match the experimental implementation, and the use of incorrect angles can lead to significant reconstruction artifacts. ## Eq. (4)$${\theta}_{i}=\mathrm{atan}2({v}_{i}/{u}_{i}),\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\theta}_{i}\in [-\pi :\frac{\pi}{12}:\pi ),$$Our self-calibration algorithm starts with an initial guess ${u}_{i}^{\mathrm{init}},{v}_{i}^{\mathrm{init}}$ from the algorithm in Ref. 36, whose estimated LED positions are often contaminated by noise. Accordingly, the final calibrated LED positions ${u}_{i}^{\mathrm{cal}},{v}_{i}^{\mathrm{cal}}$ are parameterized as ## Eq. (5)$${u}_{i}^{\mathrm{cal}}=\mathrm{\Delta}u+{\mathrm{NA}}_{\mathrm{obj}}\text{\hspace{0.17em}}\mathrm{cos}({\theta}_{i}+\mathrm{\Delta}\theta )/\lambda ,\phantom{\rule{0ex}{0ex}}{v}_{i}^{\mathrm{cal}}=\mathrm{\Delta}v+{\mathrm{NA}}_{\mathrm{obj}}\text{\hspace{0.17em}}\mathrm{sin}({\theta}_{i}+\mathrm{\Delta}\theta )/\lambda ,$$## 3.## Results## 3.1.## Angle Self-Calibration and Performance CharacterizationAchieving high-quality RI tomographic reconstruction requires accurate LED positioning, especially when imaging large-volume objects under high NA illuminations. In practice, removing all residual errors in the LED positions using only manual alignment and physical calibration procedures is nontrivial. We instead develop an algorithmic self-calibration method for finely tuning the LED positions and demonstrate our technique’s improvement of 3-D reconstructions. Our self-calibration method combines two main principles for high-accuracy measurements. First, our TF analysis shows that each intensity image’s Fourier spectrum should contain distinct circular regions with center positions defining the illumination angle. A demonstration of this principle on the experimental data is shown in Video 1. A previously developed algorithm We demonstrate the effectiveness of this method on diatom microalgae (S68786, Fisher Scientific) fixed in glycerin gelatin imaged with a 0.65 NA MO. An example intensity image is shown in Fig. 2(a). The low-absorbing features (i.e., “phase” features) are already visible due to asymmetric illumination, akin to differential phase contrast. As shown in Fig. 2(d), the LED miscalibrations have minimal effect for structure reconstructions at the objective’s focal plane ($z=0\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$). Significant RI degradation from incorrect illumination angles is observed at defocus reconstruction planes [Figs. 2(c) and 2(e)]. This degradation is intuitively explained under the “light field” effect: ## 3.2.## Tomographic Characterization of |

*in vitro*intensity diffraction tomography," Advanced Photonics 1(6), 066004 (28 December 2019). https://doi.org/10.1117/1.AP.1.6.066004