Draft:Rotating coherent scattering microscopy

Theory of image formation

A ROCS image is obtained by ${\displaystyle \phi }$ = 0...2π azimuthal rotation and integration of coherent intensity images acquired for each illumination direction. Neglecting polarizations effects, the angle-dependent interference between the reflected field ${\displaystyle E_{r}(\mathbf {r} ,\phi )}$​ and the backscattered field ${\displaystyle E_{s}(\mathbf {r} ,\phi )}$​ yields

${\displaystyle I_{\mathrm {ROCS} }(\mathbf {r} )==\int _{0}^{2\pi }\left|E_{r}(\mathbf {r} ,\phi )+E_{\mathrm {s} }(\mathbf {r} ,\phi )\right|^{2}d\phi }$

The ROCS image intensity can be decomposed into three contributions: a background term, a dark-field scattering term, and an interferometric enhancement term. Explicitly,

${\displaystyle I_{\mathrm {ROCS} }(\mathbf {r} )==\int _{0}^{2\pi }\left(I_{r}(\mathbf {r} ,\phi )+I_{s}(\mathbf {r} ,\phi )+I_{inf}(\mathbf {r} ,\phi )\right)\,d\phi }$

with background intensity ${\displaystyle \int I_{r}d\phi }$ , with the darkfield image ${\displaystyle I_{\mathrm {ROCS} }^{\mathrm {DF} }(\mathbf {r} )=\int I_{s}(\mathbf {r} ,\phi )d\phi }$

and with an interferometric enhancement term, controlled by the degree of coherence ${\displaystyle \gamma }$.^[2]

${\displaystyle I_{\mathrm {ROCS} }^{\mathrm {inf} }(\mathbf {r} )=2\gamma \int {\sqrt {I_{r}(\mathbf {r} ,\phi )\,I_{s}(\mathbf {r} ,\phi )}}\sin {\bigl (}\Delta \varphi (\mathbf {r} ,\phi ){\bigr )}\,d\phi }$

For a distribution of N scatterers, the object function is ${\displaystyle f(\mathbf {r} )=\sum \nolimits _{j=1}^{N}f_{j}(\mathbf {r} -\mathbf {r} _{j})}$ and the backscattered fields at the camera are ${\displaystyle E_{cam}(\mathbf {r} ,\phi )==\sum \nolimits _{j}E_{cam,j}(\mathbf {r} ,\phi )==\sum \nolimits _{j}|E_{sj}(\mathbf {r} ,\phi )|\,e^{i\varphi _{j}(\mathbf {r} ,\phi )}}$.

Interferometric amplification in darkfield ROCS

Also dark-field ROCS images exhibit enhanced spatial resolution and contrast due to the mutual interference of the scattered fields originating from multiple particles. For N scattering particles the scattered fields ${\displaystyle |E_{sj}|\,e^{i\varphi _{j}}}$ from individual particles interfere coherently at the camera plane (with index j < N).

The dark-field ROCS intensity is therefore given by

${\displaystyle I_{\mathrm {ROCS} }^{\mathrm {DF} }(\mathbf {r} )==\int _{0}^{2\pi }\left|\sum \nolimits _{j}|E_{sj}(\mathbf {r} ,\phi )|\,e^{i\varphi _{j}(\mathbf {r} ,\phi )}\right|^{2}}$${\displaystyle ==\int _{0}^{2\pi }\sum \nolimits _{j}I_{\mathrm {cam} ,j}(\mathbf {r} ,\phi )\,d\phi +\gamma \int _{0}^{2\pi }\sum \nolimits _{j\neq k}{\sqrt {I_{\mathrm {cam} ,j}\,I_{\mathrm {cam} ,k}}}\cos \!{\bigl (}\Delta \varphi _{jk}(\mathbf {r} ,\phi ){\bigr )}\,d\phi }$

The first term represents the angular integration of the N self-interference (intensity) contributions from individual particles. The second term contains N(N−1) cross-interference contributions, weighted by the degree of coherence γ between the scattered fields^[2]:

Increase in spatial resolution

The increase in spatial resolution results from the oblique illumination of the object by a plane wave, ${\displaystyle e^{-i\,\mathbf {k} _{i}(\phi ,\theta )\cdot \mathbf {r} }\cdot f(\mathbf {r} )}$ with ${\displaystyle \theta }$ being the polar angle of illumination. In Fourier space, the object spectrum is shifted, ${\displaystyle F(\mathbf {k} -\mathbf {k} _{i}(\phi ,\theta ))}$ enabling transfers of higher spatial frequencies of the fields backscattered from the object. The ROCS image intensity reads

${\displaystyle I_{\mathrm {ROCS} }(\mathbf {r} ,q)=\int _{0}^{2\pi }\left|\left[(q+if(\mathbf {r} ))e^{-i\mathbf {k} _{r}(\phi ,\theta )\cdot \mathbf {r} }\right]\star h_{c}(\mathbf {r} )\right|^{2}\,d\phi }$

Here ${\displaystyle h_{c}(\mathbf {r} )}$ denotes the coherent point spread function (PSF) and the ${\displaystyle \star }$ - symbol a convolution, prior to modulus squaring. The parameter q distinguishes between darkfield ${\displaystyle I_{\mathrm {ROCS} }^{\mathrm {DF} }=I_{\mathrm {ROCS} }(q=0)}$ and brightfield ${\displaystyle I_{\mathrm {ROCS} }^{\mathrm {BF} }=I_{\mathrm {ROCS} }(q=1)}$.

ROCS microscopy is based on field modulation and can be considered the coherent counterpart of structured illumination microscopy (SIM), which relies on intensity modulation. In both techniques, oblique illumination is used to generate spatial modulation of the object illumination, resulting in shifts of the object spectrum in Fourier space.

Speckles

Angular rotation does not average out speckles. The speckle patterns observed under illumination from a single direction represent direction-dependent distortions of the object structure. As the illumination rotates and the object is probed from multiple azimuthal directions, these distorted patterns combine to reconstruct the correctly shaped object features in the integrated backscattered image.