## Abstract

Small structures, as inside living cells, move on millisecond timescales, which is usually far beyond the imaging rate of superresolution fluorescence microscopes. In contrast, label-free imaging techniques providing high photon densities can operate at $>100\text{\hspace{0.17em}}\mathrm{Hz}$. For simple structures, an oblique, coherent illumination with a static laser beam increases image contrast and resolution considerably, whereas illumination of complex structures results in an image full of speckles. Remarkably, an artifact-free image is generated by subsequent oblique illumination of the structure from all azimuthal directions. This is the working principle of ROCS microscopy, which currently achieves 150 nm spatial and 10 ms temporal resolution without fluorophore bleaching, and is therefore highly beneficial for live-cell imaging. However, the complicated formation of ROCS images and image spectra during one sweep, i.e., the superposition of different speckle patterns is still unclear. Here, we investigate with experiments and computer simulations the influence of speckle-like interference patterns on the final image contrast and resolution, in darkfield mode and, by adding a reference wave, in brightfield mode. In close comparison to experimental results, we present a theoretical framework, which describes the ROCS image formation in real space and in $k$ space by identifying different spectral components. In addition, we vary the degree of coherence by a rotating diffuser and thereby demonstrate that maximal spatial coherence and maximal speckle interference from multiple scattering provide the best image contrast and resolution. We find that the cross correlations of elementary waves emitted in a distance of several micrometers to each other positively contribute to image formation and do not, as commonly believed, distort image formation. By understanding the composition of image speckles in time and space, future coherent microscopes should provide new insights into the high-speed world of living cells.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Teamwork usually provides better results and higher operation efficiency, because one player can benefit from the action of the other. This is also true for coherent photons, which can amplify the other photons’ electric field and thereby the efficiency of signal transfer. In microscopy, such correlated photons illuminate an object, scatter nonresonantly and propagate to the detector to form a coherent image by interfering with each other. Although incoherently operating photons have provided impressive imaging results through advanced photophysical principles [1,2], they form an image independently of each other, such that image contrast is often limited by the number of photons reaching the camera. Coherent photons, on the other side, can enable superior image contrast through constructive and destructive interference with a much lower number of photons and therefore a much shorter camera integration time [3].

Super(-ior) spatial resolution is already achieved, when objects in a distance of a quarter of the wavelength or less can be separated in an image. The idea is to image two adjacent points not at the same time on a detector, but to image one object point after the other, which can be only a few tens of nanometers away. This can be realized in a stochastic manner as by the superresolution microscopy techniques in PALM & STORM [4] or by a targeted manner as in STED or RESOLFT [5]. Even in structured illumination microscopy (SIM) [6], one thin line of intensity excites one object point, whereas an adjacent object point remains dark, before the intensity line pattern is shifted to illuminate only the second object point [7].

In a comparable way, a twofold increase in spatial resolution can be achieved by oblique, coherent illumination of an object [8,9]. In consequence, two adjacent objects oriented along the illumination axis are illuminated one after the other, such that one coherently scattered spherical wave reaches the detector phase-delayed relative to the second spherical wave. This results in destructive interference at the camera and an intensity separation of two objects in a distance of $d\approx \lambda /4$. This super(-ior) resolution has been exploited in various arrangements in synthetic aperture microscopy [10–14].

The usual problem is that the wave scattered coherently at an arbitrary arrangement of scatterers forms an image full of speckles at the camera, which has not much in common with a comparable incoherent (fluorescence) image. However, by illuminating the distribution of scatterers subsequently from all directions, two adjacent scatterers are always in line with at least two illumination directions, thereby providing not only a strong separation contrast, but also a nearly artifact free image [11].

We have termed this technique rotating coherent scattering (ROCS) microscopy. Here, a $2\pi $ azimuthal scan of an obliquely incident, collimated laser beam is performed during the integration time of a camera to generate a partially coherent image, which reveals superior contrast and a resolution of about 150 nm without any postprocessing [15]. ROCS enables image acquisitions of living, highly active cells at 50–100 Hz, a frame rate that is limited only by the scan-mirror or the camera. Without requiring fluorophores or other labels, thousands of images could be monitored online without any bleaching or loss in image quality, opening new insights into the unexpectedly high dynamics of cellular actin structures [16], of deforming bacteria or the surprising viscoelastic behavior of single biofilaments like microtubules [17]. Here it was shown that ROCS works both in darkfield (DF) and brightfield (BF) modes and both in total internal reflection (TIR) and non-TIR mode.

Figure 1 shows two TIR images of the same living cell (mouse bone marrow-derived macrophage) [18] with tiny cellular structures like branches of thin filopodia. ROCS (in DF mode) provides significantly higher image contrast and resolution [Fig. 1(a), 9 ms integration time] than fluorescence, when suitable labels such as LifeAct GFP-actin are used [Fig. 1(b), 200 ms integration time]. The strong optical transfer of the image spectra (Fourier transforms) at high spatial frequencies $>6/\mathrm{\mu m}$, but also the high image contrast opens questions about the image forming process and the possibilities to further improve this imaging technique.

The purpose of this paper is to enlighten the principles of ROCS imaging both in DF and BF modes. This is performed through a side-by-side comparison between experiments and imaging simulations of simple arrangements of beads, together with a detailed theoretical description of image formation in real space and $k$ space.

## 2. PRINCIPLES OF A ROCS MICROSCOPE

The working principles of the ROCS microscope used in this study are illustrated in Fig. 2. A fiber coupled 405 nm diode laser (Oxxius LBX-405, with linewidth $\mathrm{\Delta}\lambda \approx 1.5\text{\hspace{0.17em}}\mathrm{nm}$) is expanded and deflected by a 2D scan mirror (SM, Sunny Technologies, S8210M). A lens focuses the collimated wave (shown in light blue) onto a rotating diffuser, which can be placed optionally in the beam path to change the degree of coherence $\gamma $. Another 4f system focuses the beam via a 50:50 beam splitter (BS, pellicle) into the back focal plane (BFP) of the objective lens, such that the laser focus rotates in the TIR ring, when the SM is switched on (azimuthal $2\pi $ scan). Particles (Polybeads, Polysciences Inc.) adherent at the coverslip (CS) are obliquely illuminated with a polar angle of $\theta \approx 70\xb0$ using a high-NA objective lens (Leica HCX PL Apo, $100\times $, NA 1.46, oil). Coherently scattered laser light (shown in midblue) is projected by the objective, the tube lens and a further 4f system onto a CMOS camera (Point Grey GS3- U3-23S6M-C). The illumination beam is reflected at the upper interface of the coverslip and is guided through a 6f system to a Fourier plane, where the rotating laser focus can be blocked by a DF stop, reducing the effective detection ${\mathrm{NA}}_{\mathrm{det}}$ to about $p=80\%$. This diaphragm can be fully opened to enable BF ROCS, where the back reflected, unscattered light interferes with scattered light at the camera.

## 3. THEORY OF ROCS IMAGE FORMATION

#### A. Object Distribution

We assume a distribution of polarizable objects $\sum _{j}{f}_{j}$ with refractive indices ${n}_{j}$ different than the aqueous medium around them with ${n}_{m}\approx 1.33$. Each 3D object ${f}_{j}(\mathbf{r}-{\mathbf{r}}_{j})$ (with particle index $j$) is defined by a shape function ${s}_{j}(\mathbf{r})$, a polarizability ${\alpha}_{j}$, and its position ${\mathbf{r}}_{j}$. Therefore, the object function and the corresponding object spectrum $F(\mathbf{k})=\mathrm{FT}[f(\mathbf{r})]$ as the Fourier transform of $f(\mathbf{r})$ can be written as

The wave vector ${\mathbf{k}}_{\mathbf{i}}(\varphi ,\theta )=\frac{2\pi}{\lambda}{n}_{i}(\mathrm{sin}\text{\hspace{0.17em}}\varphi \xb7\mathrm{sin}\text{\hspace{0.17em}}\theta ,\mathrm{cos}\text{\hspace{0.17em}}\varphi \xb7\mathrm{sin}\text{\hspace{0.17em}}\theta ,\mathrm{cos}\text{\hspace{0.17em}}\theta )$ of the incident plane wave can be changed in angle by the scan mirror, which shifts the focus position in the BFP. While the polar illumination angle $\theta $ is typically kept constant, the rotating azimuthal angle $\varphi $ is varied over the full $2\pi $ range within a few milliseconds. The refractive index of the immersion medium is ${n}_{i}\approx 1.52$ and the laser wavelength is $\lambda =405\text{\hspace{0.17em}}\mathrm{nm}$. The polar angle $\theta \le {\mathrm{sin}}^{-1}({\mathrm{NA}}_{\mathrm{ill}}/{n}_{i})=74\xb0$ is limited by the illumination numerical aperture ${\mathrm{NA}}_{\mathrm{ill}}=1.46$.

#### B. Spatial Coherence

The phase of the incident plane wave can be altered by an angle $\u03f5$ dependent phase distortion $\phi (\u03f5)=\delta \mathbf{k}(\u03f5)\xb7\mathbf{r}$ generated by the rotating diffuser wheel. The rotating wheel generates a spatially incoherent image by adding illumination intensities one after the other from different directions, effectively broadening the illumination spectrum. A single $\phi $-distorted, incident (plane) wave can be expressed by

#### C. Scattered Fields

For simplicity, we use the scalar Born approximation to calculate the scattered field

#### D. Camera Image

For coherent illumination ($\gamma (\u03f5=0)=1$) the resulting SD image intensity at the camera is simply the modulus square of the electric field, ${I}_{\mathrm{SD}}(\mathbf{r},\varphi ,q,0)={|{E}_{\mathrm{cam}}(\mathbf{r},\varphi ,q,0)|}^{2}={|[(q+f(\mathbf{r})*g(\mathbf{r}))\xb7{E}_{i}(\mathbf{r},\varphi ,0)]*{h}_{c}(\mathbf{r})|}^{2}$. For partial coherence ($\gamma (\u03f5)<1$), the $\u03f5$-angular autocorrelation of the electric field, ${I}_{\mathrm{SD}}(\mathbf{r},\u03f5)=\frac{1}{2{\theta}_{0}}{\int}_{-{\theta}_{0}}^{{\theta}_{0}}{E}_{\mathrm{cam}}(\mathbf{r},\theta +\u03f5)\xb7{E}_{\mathrm{cam}}^{*}(\mathbf{r},\theta )\mathrm{d}\theta $ describes the image intensity. It depends on the spatial (radial) coherence defined by the small polar angular variation ϵ and splits into three terms (see Supplement 1):

The second term, the darkfield intensity ${I}_{\mathrm{SD}}^{\mathrm{DF}}(\mathbf{r},\varphi ,\u03f5)$, can be split also into a distribution of scattered waves with index $j$:

#### E. ROCS Image

In ROCS microscopy, the final image at the camera $I(\mathbf{r},\theta ,q,\u03f5)={\int}_{0}^{2\pi}{I}_{\mathrm{SD}}(\mathbf{r},\varphi ,q,\u03f5)\mathrm{d}\varphi $ is obtained after incoherent addition of coherent images, i.e., after $\varphi $-angular integration of intensities:

The second line of Eq. (12) is illustrated by Fig. 1(a), which shows the ROCS image $I(\mathbf{r},q=0,\u03f5)$ of a living cell, recorded in DF mode ($q=0$). The numerical apertures ${\mathrm{NA}}_{ill}=1.42$ and ${\mathrm{NA}}_{\mathrm{det}}=1.17$ were adjusted according to a standard calibration procedure.

In the simulation results of Fig. 3, the split of the DF ROCS image into an incoherent image of purely scattered light and the multiple interference pattern is demonstrated (analytical solution of integrals in Eq. (9) in the case of DF-ROCS using Mathematica, Wolfram Research). The interference pattern provides additional contrast and resolution, by adding negative dips (shown in blue) between adjacent bead images. In most cases, the interference intensity is zero at the position of a bead, as indicated by the position of the black circles in Fig. 3(c).

In several cases, the composition of ROCS images can be better understood by analyzing the image spectra in Fourier space than by looking at local destructive interferences of scattered waves.

#### F. Field Spectrum

The electric field spectrum at the camera is obtained by a Fourier transform of Eq. (8) with $G(\mathbf{k})=\mathrm{FT}[g(\mathbf{r})]=i\xb7\delta ({k}_{z}\pm \sqrt{{n}_{i}^{2}{k}_{0}^{2}-{k}_{\perp}^{2}})/\sqrt{{n}_{i}^{2}{k}_{0}^{2}-{k}_{\perp}^{2}}$ being the Weyl form of the Ewald sphere [19]. ${H}_{c}(\mathbf{k})=\delta ({k}_{z}\pm \sqrt{{n}_{i}^{2}{k}_{0}^{2}-{k}_{\perp}^{2}})/\sqrt{{n}_{i}^{2}{k}_{0}^{2}-{k}_{\perp}^{2}}\xb7\mathrm{circ}\left(\frac{{k}_{\perp}}{p\xb7{k}_{d}}\right)$ is the reflection coherent transfer function, which is an Ewald spherical cap with cap radius $p\xb7{k}_{d}=p\xb7{k}_{0}\xb7{\mathrm{NA}}_{\mathrm{det}}<{k}_{0}\xb7{n}_{i}$ [Fig. 4(c)]. Therefore, the twofold low-pass filtering of the scattered field in Eq. (8) reduces to $G(\mathbf{k})\xb7{H}_{c}(\mathbf{k})=i\xb7{H}_{c}(\mathbf{k})$. Disregarding the degree of coherence for a moment, the 3D field spectrum for a single illumination direction ${\mathbf{k}}_{i}(\varphi )$ reads

According to the Born approximation, the reflected field and the object spectrum are shifted by the incident wave vector ${\mathbf{k}}_{i}(\varphi )$ and multiplied with the Ewald spherical cap ${H}_{c}(\mathbf{k})$, resulting in a narrow axial field spectrum ${\tilde{E}}_{\mathrm{cam}}({k}_{z},\varphi )$ or extended axial electric field ${E}_{\mathrm{cam}}(z,\varphi )$, respectively. For an approximately two-dimensional object or for an evanescent wave illumination, Eq. (13) describes a two-dimensional image formation process equally well. Hence, the spherical cap can be assumed to be a circular disk ${H}_{c}(\mathbf{k})\approx \mathrm{circ}(\frac{k}{p\xb7{k}_{d}})$ after projection to the plane $z=0$ (i.e., ${k}_{z}\approx \text{const}$).

The image formation process is illustrated by Fig. 4(c) for the case the detection angle is larger than the illumination angle, ${\mathrm{NA}}_{\mathrm{det}}>{\mathrm{NA}}_{\mathrm{ill}}$. The object spectrum is shown in gray; the coherent transfer function is shown in red. Although the wave vectors for illumination and detection have the same length $2\pi /\lambda $ in 3D, the $z$ projection of the maximal angle scattered $k$ vector ${\mathbf{k}}_{s}$ is longer than the $z$ projection of ${\mathbf{k}}_{i}$.

#### G. Single-Direction Intensity Spectrum

The spectrum of an intensity image ${\tilde{I}}_{\mathrm{SD}}({\mathbf{k}}^{\prime},\varphi ,\tilde{q})={\mathrm{FT}}_{xy}[{I}_{\mathrm{SD}}(\mathbf{r},\varphi ,q)]$ illuminated from a single direction $\varphi $, is obtained by the autocorrelation $\mathrm{AC}(\mathbf{k})$ of the electric field spectrum ${\tilde{E}}_{\mathrm{cam}}(\mathbf{k},\varphi )$

In the Supplement 1, all four AC integrals ${\tilde{I}}_{1\mathrm{SD}}({\mathbf{k}}^{\prime})\dots {\tilde{I}}_{4\mathrm{SD}}({\mathbf{k}}^{\prime})$ are solved to identify their role in ROCS image formation. The first term ${\tilde{I}}_{1\mathrm{SD}}({\mathbf{k}}^{\prime})$ with the Kronecker delta function $\delta (\mathbf{k}-{\mathbf{k}}_{i})$ does not contain any object information, whereas the second term ${\tilde{I}}_{2\mathrm{SD}}({\mathbf{k}}^{\prime})$ provides the darkfield (DF) image spectrum ${\tilde{I}}_{\mathrm{SD}}^{\mathrm{DF}}({\mathbf{k}}^{\prime})$ with access to high object frequencies.

#### H. Interference Spectrum

The sum of the last two spectra, ${\tilde{I}}_{3\mathrm{SD}}(\mathbf{k},\varphi ,\tilde{q})+{\tilde{I}}_{4\mathrm{SD}}(\mathbf{k},\varphi ,\tilde{q})$, represents the single direction (SD) interference spectrum. It is characterized by shifts of the transfer function in the illumination direction and opposite to it, as sketched by the blue vectors in Figs. 4(b) and 4(d). ${H}_{c}({\mathbf{k}}^{\prime}+{\mathbf{k}}_{i})$ and ${H}_{c}({\mathbf{k}}^{\prime}-{\mathbf{k}}_{i})$ extract high object frequencies as well.

As derived in the supplementary information, the degree of coherence $\gamma (\u03f5)$ weights the contribution of this interference spectrum:

While the shift of the two coherent transfer functions of the interference spectrum is proportional to the lateral component of the incident $k$ vector, the diameter of the darkfield spectrum is only determined by the detection numerical aperture ${\mathrm{NA}}_{\mathrm{det}}$ [see experimental results in Fig. 4(e)].

#### I. ROCS Image Spectrum

Hence, our final ROCS image spectrum $\tilde{I}({\mathbf{k}}^{\prime},\tilde{q})={\mathrm{FT}}_{xy}[I(\mathbf{r},q)]$ after angular integration at the camera reads

The ROCS darkfield spectrum ${\tilde{I}}^{\mathrm{DF}}(\mathbf{k},\u03f5)={\int}_{0}^{2\pi}{\tilde{I}}_{\mathrm{SD}}(\mathbf{k},\varphi ,0,\u03f5)\mathrm{d}\varphi $, as shown in Fig. 1(a) can be written as

In general and especially for experiments, the integrated interference spectrum ${\tilde{I}}_{\mathrm{inf}}(\mathbf{k},\tilde{q})=\tilde{I}(\mathbf{k},\tilde{q})-\tilde{I}(\mathbf{k},0)-A\delta (\mathbf{k})$ can be obtained by subtracting the darkfield spectrum and the central delta peak from the total (brightfield) spectrum. In this case, an effective, shift invariant transfer function can be obtained, enabling processes such as image deconvolution (see Supplement 1).

Figure 5 illustrates the stepwise ROCS image formation for a simple object consisting of three straight lines through a numerical simulation based on the above equations (using MATLAB, Mathworks). For three exemplary illumination directions, the resulting electric fields at the camera are shown [Fig. 5(a)], together with the corresponding phases [Fig. 5(b)] and intensities [Fig. 5(c)]. Figure 5(d) illustrates the composition of the single direction field spectra and the corresponding autocorrelations. Figure 5(e) displays the three-line object and the final darkfield image with the corresponding Fourier spectra, consisting of three lines as well.

## 4. RESULTS

The results presented in the following figures are selected such that they can be well compared with the numerical simulations (programmed in MATLAB, Mathworks) and the analytical calculations to understand ROCS image formation in detail and to set the basis for further improvements in image quality in the future.

#### A. Temporal Coherence and Spectral Shape

For a given and constant polar angle of illumination $\theta $, Eq. (9) describes the SD image intensity ${I}_{\mathrm{SD}}(\mathbf{r},\varphi ,q,\u03f5)$ depending on the azimuthal direction $\varphi $ of the incident plane wave, the fraction $q$ of unscattered light, and the spatial coherence defined by $\u03f5$. Consequently and according to Eq. (12), the final ROCS image $I(\mathbf{r},q,\u03f5)$ integrated over all directions (AD) does not show any dependence on the azimuthal direction $\varphi $.

This is demonstrated in Fig. 6 for a distribution of 200 nm polystyrene spheres (Polybeads, Polysciences, Inc.), where the SD images in the second column reveal interference fringes and artifacts (see Visualization 1 and Visualization 2), which are characteristic for the plane wave’s azimuthal direction of incidence $\varphi $ indicated by the white arrow. The ROCS images are obtained by angular integration over all directions $\varphi $ as illustrated by the stepwise composition in the Visualization 3 and Visualization 4. The final ROCS images are shown in the third column and demonstrate superior contrast and resolution without visible coherence artifacts. The corresponding Fourier transforms are displayed for the SD images in the first column and for the final AD (ROCS) images in the fourth column. In all experiments, the polar illumination angle was $\theta =70\xb0$, corresponding to ${\mathrm{NA}}_{\mathrm{ill}}=1.42$.

The three rows in Fig. 6 represent three different imaging modes. Figure 6(a) shows a pair of brightfield images (SD and AD) with reduced temporal coherence, ${I}^{\mathrm{BF}}(\mathbf{r},\u03f5)=I(\mathbf{r},q\approx 1,\u03f5)$, where the rotating diffuser wheel generated a variation of illumination modes within the polar angle range $|\u03f5|\le {\theta}_{0}$, decreasing the coherence significantly. Effectively, the pupil point illumination and the illumination ring after a $2\pi $ azimuthal sweep are broadened. This becomes apparent by the less pronounced interference fringes on the left side of the particles visible in the SD images.

After removing the diffuser wheel [Fig. 6(b)], the SD image ${I}^{\mathrm{BF}}(\mathbf{r},\u03f5\approx 0)=I(\mathbf{r},q\approx 1,\u03f5\approx 0)$ of the same distribution $f(\mathbf{r})$ of spheres shows significantly stronger interference fringes, although the final ROCS image in the third column looks similar. However, the contrast increases with the coherence as indicated by three beads in a row highlighted with colored rectangles and line scans plotted next to it. The effect of the temporal coherence is also well visible in the Fourier transforms, i.e., the image spectra ${\tilde{I}}_{\mathrm{SD}}(\mathbf{k},\varphi ,\tilde{q})$ and $\tilde{I}(\mathbf{k},\tilde{q})$ in the 1st and 4th columns: in the case of reduced temporal coherence, the spectra are damped especially at higher frequencies, whereas an enhanced transfer of high frequencies is visible for illumination with high coherence. This is further quantified by the white line scans of the final ROCS image spectra.

The third row [Fig. 6(c)] shows the corresponding darkfield images, ${I}^{\mathrm{DF}}(\mathbf{r},0)=I(\mathbf{r},q=0,0)$, after blocking the zeroth order by the diaphragm. By this DF stop, the effective detection ${\mathrm{NA}}_{\mathrm{det}}$ was reduced to about $p=80\%$. In consequence, the image contrast increased strongly, but the transfer of higher spatial frequencies was cut off, which is well visible in the two image spectra. Again, artifacts from scattered wave interferences distort the SD image, whereas all artifacts disappear in the corresponding ROCS image (AD).

Because it requires careful alignment to ensure the same intensity from all illumination directions, we took some (more or less) isolated beads from Fig. 6 for a further analysis in Figs. 7 and 8, to ensure the comparability.

#### B. Two Bead Images in Darkfield and Brightfield Modes

In a next investigation step, we reduced the complexity of the interferences from many scattered waves. Hence, for the simplest case of two adjacent beads with a diameter $2a=200\text{\hspace{0.17em}}\mathrm{nm}$, we measured and simulated the resulting SD and AD images at the camera with maximal coherence as shown in Figs. 7(a) and 7(b). The corresponding image spectra (Fourier space) are shown below in Figs. 7(c) and 7(d). In all four figures, the respective top row represents the DF mode (${\mathrm{NA}}_{\mathrm{det}}=1.20$) and the respective bottom row the BF mode (${\mathrm{NA}}_{\mathrm{det}}=1.46$). The final ROCS images in the first column of Figs. 7(a) and 7(b) demonstrate the resolution and contrast in the images of 200 nm beads. The respective second column displays the bead images from a single illumination direction nearly parallel to the axis of the two beads (SD, $\varphi \approx -10\xb0$), revealing two distorted spots as bead images, which are well (DF) or less well (BF) resolvable. The respective third column displays the images of the two beads from a nearly orthogonal direction (SD, $\varphi \approx 80\xb0$), which cannot be resolved, because the nearly spherical waves interfere at the camera without relative phase shift. Although the distance of about $2a=200\text{\hspace{0.17em}}\mathrm{nm}$ between the bead centers is constant, the distance $\mathrm{\Delta}{x}_{\mathrm{SD}}(\varphi )$ of the bead images with parallel illumination is increased, but decreased with orthogonal illumination. The final ROCS image generates an angularly averaged distance $\mathrm{\Delta}{X}_{\mathrm{AD}}$, which is about 10% longer than the actual contact distance of 2a.

The image spectra in Figs. 7(c) and 7(d) represent the results expressed by Eqs. (14) for SD and (17) for AD both for the DF mode ($q=0$) and BF mode ($q\approx 1$). The respective ROCS optical transfer limits the spatial frequencies given by the object spectrum ${F}_{2P}(\mathbf{k})=2{F}_{0}(\mathbf{k})\xb7\mathrm{cos}({k}_{x}a)$ in Eq. (3), which is a (spherical) Bessel function ${F}_{0}(\mathbf{k})$ modulated with $\mathrm{cos}({k}_{x}a)$ and resembles the well-known diffraction pattern of a double pinhole. Despite the logarithmic representation, this fringe pattern spectrum is well visible in the final ROCS image spectra of the respective first row of Figs. 7(c) and 7(d). Both in experiment and simulation, the SD darkfield spectra ${\tilde{I}}_{\mathrm{SD}}^{\mathrm{DF}}({\mathbf{k}}^{\prime})\propto AC[2{F}_{0}(\mathbf{k})\xb7\mathrm{cos}({k}_{x}a)\xb7{H}_{c}(\mathbf{k}+{\mathbf{k}}_{i})]$ show a vertical fringe pattern, clearly visible in nearly parallel illumination direction ($\varphi \approx -10\xb0$, second row), and dimly in the nearly orthogonal direction ($\varphi \approx 80\xb0$, third row). Remarkably, the spectral fringe distance $\mathrm{\Delta}{k}_{SD}(\varphi )\ne \frac{2\pi}{a}$ is minimal for parallel illumination (i.e., a maximal distance of the two spots in real space) and is maximal for orthogonal illumination (i.e., a minimal spot distance in real space). Correspondingly, the ROCS image spectrum reveals a spectral fringe distance $\mathrm{\Delta}{k}_{\mathrm{AD}}$ averaged over all illumination directions. Besides the central data peak, the brightfield image spectra additionally reveal the double ring-like interference spectrum, which is clearly visible in both directions in the simulation. In the experiment, the signal-to-noise ratio is rather weak because of the small amount of light scattered at the two 200 nm beads within the integration time of 9 ms. At least in the nearly parallel illumination direction ($\varphi \approx -10\xb0$), the ring-like structure of the interference spectrum is visible. As expected, the spectrum width in the BF mode (${\mathrm{NA}}_{\mathrm{det}}=1.46$) is broader than in the DF mode (${\mathrm{NA}}_{\mathrm{det}}=1.20$) as indicated by the dashed circles in Fig. 7(c). The image composition and the structure of orthogonal SD images, especially in the BF mode are further illustrated in the supplementary information (Figs. S2 and S3).

#### C. Three Bead Images in Darkfield and Brightfield Modes

In the same way as for the images of the two beads, we measured, simulated, and analyzed an arrangement of three beads with distances $|{\mathbf{d}}_{ij}|>1\text{\hspace{0.17em}}\mathrm{\mu m}$ to each other. As described by Eq. (4) and visible in Fig. 8, the triangular bead arrangement with distance vectors ${\mathbf{d}}_{ij}$ in real space (top row) result in an image spectrum with triangular symmetry defined by the reciprocal axis ${\mathbf{s}}_{ij}$. perpendicular to ${\mathbf{d}}_{ij}$. Similar to crystallography, the reciprocal grating, i.e., the object spectrum, consists of basic cells with dimensions ${|{\mathbf{d}}_{ij}|}^{-1}$, which are well visible in logarithmic scaling both in the experiment [Fig. 8(a)] and in the simulation [Fig. 8(b)]. In the left of Fig. 8(a), two SD images of three beads in BF are depicted together with their corresponding spectra, which show a pronounced double ring like the interference spectrum. Again, it is observed that the structure of the object spectrum ${|{F}_{3P}(\mathbf{k})|}^{2}$, although cut off at $2{k}_{0}{\mathrm{NA}}_{\mathrm{det}}$ and weighted by $\mathrm{AC}[{H}_{c}(\mathbf{k}-{\mathbf{k}}_{i})]$, remains invariant upon illumination from different directions, such that the incoherent superposition of many spectra do not alter the reciprocal grating. The simulated spectra reveal a dark ring, which is the zero intensity of the spherical Bessel function of a 200 nm bead and which is clearly inside the optical ROCS transfer. This minor effect is not visible in the experimental data, which is too noisy because of the small signals of only three 200 nm beads.

## 5. DISCUSSION

ROCS microscopy provides high contrast images at a spatial resolution of 150 nm (at 490 nm wavelength) [11,15] within millisecond acquisition times, which is especially beneficial for dynamic systems such as live cell imaging [see image with 9 ms acquisition time in Fig. 1(a)]. No fluorophores and no image postprocessing are necessary.

Image generation in ROCS microscopy bases on the incoherent superposition of coherent images from different illumination angles. Each coherent image consists of a complicated interference pattern on the camera generated by many partial waves scattered at object structures in the sample plane. Although the final ROCS image is nearly free of artifacts, a single direction raw image is full of interference fringes and speckles, and its analysis is complicated, if not impossible. The image spectrum (the Fourier transform), however, reveals information, which can be better interpreted, especially in brightfield mode, where additional interference with the unscattered light is enabled. The goal of this study has been to shed light on the image formation process and to decouple the components of unscattered light, purely scattered light, and interference patterns. Only with the advanced understanding of image formation, future improvements of the ROCS technique are possible.

#### A. Coherence

One of the primary questions is the role of laser illumination, where the high degree of coherence provides the basis for high interference contrast. The laser light, i.e., the wave trains emitted by our type of lasers, have typical coherence times of ${\tau}_{c}=10\text{\hspace{0.17em}}\mathrm{ps}$ and coherence lengths ${L}_{c}$ of some centimeters. Hence, temporal coherence plays a negligible role as long as ${L}_{c}$ is much larger than the particle distance. Our rotating diffuser, reduces the spatial coherence at each azimuthal illumination point $\varphi $ by averaging (autocorrelating) over the scattered fields resulting from different polar illumination directions $\theta \pm \u03f5$. The phase delay between two scattered waves from fixed scatterers illuminated under different angles results in different interference patterns at the camera and after averaging to a different interference contrast, expressed by the degree of coherence $\gamma (\u03f5)$. In traditional darkfield microscopy with annular white light illumination, interferences are suppressed by averaging: (1) different polar illumination angles, (2) azimuthal illumination angles, and (3) different wavelengths. In the temporally and radial spatially coherent case of ROCS microscopy, every partial wave from each scatterer interferes with all other partial waves, generating a complicated interference pattern [see Fig. 3(c)] and thereby contributing to the high quality ROCS image. The high correlation between photons hitting the camera over the full field of view is different for each illumination direction, but contributes to the final image after incoherent superposition. In our opinion, this is an amazing and nonintuitive effect, although it bases on the principles of coherence theory [20]. Here, the residual coherence function is defined by the Fourier transform of the effective light source intensity. An annular source with reduced ring thickness broadens the coherence function, and thereby increases the correlation between neighboring object points, or, in other words, it increases the transfer of high object frequencies. As illustrated by Fig. 6(b), the high radial spatial coherence leads to a significantly broader image spectrum (both for SD and AD illumination) through addition of the interference spectrum, which is weighted with $\gamma (\u03f5)$. Hence, $\gamma (\u03f5)$ leads to an increased contrast at high spatial frequencies, which is nothing else but increased spatial resolution.

#### B. Theory and Computer Simulations

We developed a theoretical concept, which we think is essential to understand the image formation in ROCS. Many of the formulas derived in this paper are illustrated by experimental results or simulation results, which coincide generally in a very satisfying manner. Although multiple scattering or higher scattering orders are not considered in the Born approximation, which we used throughout this paper, our results display well the interaction between the object and the coherent light. The arrangements of two contacting beads (Fig. 7) and of the three beads (Fig. 8) show the object spectrum $F(\mathbf{k}-{\mathbf{k}}_{i})$ as a reciprocal grating, which is shifted by the incident wave vector ${\mathbf{k}}_{i}$ of the high-angle oblique object illumination $f(\mathbf{r})\xb7{e}^{i{\mathbf{k}}_{i}\mathbf{r}}$. As predicted by theory, this shift is equivalent to the shift of the coherent transfer function for each illumination direction. Consequently, the structure of the object spectrum remains invariant upon autocorrelation and illumination rotation (see Fig. 8), which is characteristic for ROCS. Hence, object frequencies $F({\mathbf{k}}_{\mathrm{det}}+{\mathbf{k}}_{i})$ up to wave numbers of $|{\mathbf{k}}_{\mathrm{det}}+{\mathbf{k}}_{i}|\le {k}_{0}\xb7({\mathrm{NA}}_{\mathrm{det}}+{\mathrm{NA}}_{\mathrm{ill}})$ corresponding to a spatial resolution of 140 nm can be retrieved at high transfer intensities (at $\lambda =405\text{\hspace{0.17em}}\mathrm{nm}$).

#### C. Spectrum Analysis

The analysis of the simulated and measured image spectra, especially for single illumination directions reveals the composition of different components with typical shapes in $k$ space. We found that the darkfield image spectrum is round already for a single illumination direction, with a diameter of $4\xb7{k}_{0}{\mathrm{NA}}_{\mathrm{det}}$ independent of the illumination angle. The brightfield spectrum has an additional central zero-diffraction peak and an interference spectrum, consisting of two shifted coherent transfer functions with diameters $2\xb7{k}_{0}{\mathrm{NA}}_{\mathrm{det}}$. These are shifted in opposite directions by the illuminating $k$ vector—as illustrated in Fig. 4—giving access to higher object frequencies. Interestingly, the contributions of the interference spectrum can be controlled both by the parameter $q\le 1$, i.e., the adjustable fraction of unscattered reflected light, and the degree of temporal coherence $\gamma $, which we adjusted with a rotating diffuser. In the future, this should enable a further tuning of the image quality depending on the object properties. Due to the fast image acquisition and low photo-toxicity, optimization algorithms can be run efficiently before every experiment. An effect not yet considered in the theory is nearfield coupling between the particles and coupling to the higher refracting interface. The latter effect is visible in Figs. 4(d) and 6(b) by a finite ring thickness in the interference spectrum, which reflects the increased transfer of nearfields under supercritical angles through the optical system. The ring thickness is defined by the difference of the refractive indices $\mathrm{\Delta}n=1.52\u20131.33$ of the glass and of the aqueous solution, the beads are embedded in. The strength depends on the distance of the dipoles to the interface, but also on the degree of temporal coherence as outlined in Fig. 6.

#### D. Phase Problem

Oblique illumination results in a shifted coherent transfer function, which transfers spatial frequency information first on the one side of the object spectrum and, after a 180° azimuthal scan, on the other side of the object spectrum, such that the image spectra add up. Because the spectral shift to both sides does not occur at the same time (as in structured illumination microscopy SIM), but subsequently and therefore incoherently, phase information is lost. However, because the object spectra are symmetric for real objects, a phase jump only in the imaginary part of the spectrum occurs, which leads to the advantage of destructive interference between two adjacent scatterers, which is not resolvable under nonoblique illumination.

In the darkfield mode, i.e., without a reference wave, the electric fields at the camera illuminated from opposite directions are complex conjugate to each other, but the intensity images or the image spectra $\mathrm{AC}[F(\mathbf{k})\xb7{H}_{c}(\mathbf{k}\pm {\mathbf{k}}_{i})]$ are the same and simply add up. Hence, in DF mode, a half angular sweep, i.e., a 180° scan would be sufficient to obtain all image information, if scattering between the particles is negligible.

The brightfield mode produces the additional interference spectrum, $(F(\mathbf{k})\xb7{H}_{c}(\mathbf{k}+{\mathbf{k}}_{i}))-(F(-\mathbf{k})\xb7{H}_{c}(\mathbf{k}-{\mathbf{k}}_{i}))$, which selects the object spectrum on the one side and subtracts the object spectrum on the opposite side. The difference spectrum is point symmetric even after a full angular sweep and hence selects only the imaginary part of the object spectrum. This has the effect in real space that the images of the scatterers are modulated by a sine function (within the Born approximation) providing the typical interference fringes. In the future, further investigations will be necessary to recover the full spectral object information by applying phase shifts between scattered and unscattered light for each illumination direction.

#### E. Darkfield Versus Brightfield

A sufficient image contrast is essential for imaging small, typically cellular structures in biology, which is provided by (ROCS) darkfield microscopy. However, blocking the (rotating) zeroth order reduces the detection numerical aperture ${\mathrm{NA}}_{\mathrm{det}}$ by typically $p=0.8$. Although technically complicated, a rotating stop blocking this zeroth order would increase the effective ${\mathrm{NA}}_{\mathrm{det}}$ and avoid loss in spatial resolution. This problem can be addressed easier with (ROCS) brightfield microscopy, where a fraction $q\le 1$ of the light reflected off at the coverslip is collected. In BF mode, a significant background reduces the contrast, but offers increased resolution because no darkfield stop is required. In the simple case of two or three beads and for computer simulations based on the Born approximation, two opposite illumination directions cancel each other out, leaving the darkfield image with the full detection aperture ${\mathrm{NA}}_{\mathrm{det}}$ of the objective lens and a constant background (Figs. 6 and 7, Supplementary Figs. S2 and S3). For more complicated arrangement such as cells, different phase shifts of two waves propagating in opposite illumination directions result in different intensity distributions, which do not cancel each other out. This has to be further investigated by computer simulations using the beam propagation method (BPM). On the other hand, SD interference spectra or interference images, respectively, can be retrieved by subtracting the SD darkfield images from the SD brightfield images. In this way, it becomes possible to retrieve an effective transfer function and a narrowed effective point-spread function after integrating the SD interference image as shown in the supplementary information. In combination with another effective darkfield transfer function, this enables independent deconvolution approaches, because it is standard procedure for image reconstructions in structured illumination microscopy (SIM).

## 6. CONCLUSION AND OUTLOOK

ROCS microscopy enables principal, new insights into the unexplored high dynamics of living cells and of soft matter systems [15,16]. Currently, 140 nm spatial resolution and more than 100 Hz temporal resolution provide an unequaled spatiotemporal bandwidth, without postprocessing or loss in image quality after thousands of images. Natural questions arise, on whether 110 nm spatial resolution as in SIM is possible, whether image contrast can be increased even further, or how specificity can be added to ROCS imaging of structures, which are so far not distinguishable. The first promising results achieved in our lab, exploiting specific multiparameter correlations between different illumination directions, point out the possibilities of a further, significant increase in image resolution. With the present study, a big step forward in tackling these challenges and in understanding the complicated ROCS image formation was done, which was made possible by the theoretical framework, the computer simulations, and our considerations in $k$ space. Although further theoretical investigations and technical solutions will be necessary in the future, the control of free parameters will allow novel illumination and detection schemes with coherent light, thereby providing new views into the high-speed world of small living systems.

## Acknowledgment

The authors thank Dr. Wolfgang Singer and Dr. Felix Juenger for helpful discussions and comments on the manuscript. We thank Dr. Tim Lämmermann for providing labeled mouse bone marrow-derived macrophages.

See Supplement 1 for supporting content.

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