## Abstract

In-line holographic optical imaging has the unique capability of high speed imaging in three dimensions at rates limited only by the imaging rate of the camera used. In this technique the 3D data is recorded on the detector in a form of a hologram generated by diffraction between the scattered and unscattered light passing through the sample. For dilute samples of single particles or a small cluster of particles, this technique was shown to result in particle tracking with spatial positioning accuracy of a few nanometers. For dense suspension only approximate reconstruction were achieved with systematic axial positioning errors. We propose a scheme to extend accurate holographic microscopy to dense suspensions, by calibrating the Rayleigh-Sommerfeld reconstruction algorithm against Lorentz-Mie scattering theory. We perform this calibration both numerically and experimentally and define the parameter space in which accurate imaging is achieved, and in which numerical calibration holds. We demonstrate the validity of our approach by imaging two attached particles and measuring the distance between their centers with 36 nm accuracy. A difference of 50 nm in particle diameter is easily measured.

© 2013 OSA

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### Equations (10)

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(1)
$$\begin{array}{l}{\overrightarrow{E}}_{0}\left(\overrightarrow{r},z\right)={E}_{0}\left(\overrightarrow{r}\right){e}^{ikz}{\widehat{\epsilon}}_{0}\\ {\overrightarrow{E}}_{s}\left(\overrightarrow{r},z\right)={E}_{s}\left(\overrightarrow{r},z\right)\widehat{\epsilon}\left(\overrightarrow{r},z\right)\end{array}$$
(2)
$$I\left(\overrightarrow{r}\right)={\left|{\overrightarrow{E}}_{0}\left(\overrightarrow{r},z\right)+{\overrightarrow{E}}_{s}\left(\overrightarrow{r},z\right)\right|}^{2}={E}_{0}^{2}\left(\overrightarrow{r}\right)+2\mathit{Re}\left\{{E}_{0}\left(\overrightarrow{r}\right){E}_{s}\left(\overrightarrow{r},z\right){\widehat{\epsilon}}_{0}^{*}\cdot \widehat{\epsilon}\left(\overrightarrow{r},z\right)\right\}+{\left|{\overrightarrow{E}}_{s}\left(\overrightarrow{r},z\right)\right|}^{2}$$
(3)
$$b\left(\overrightarrow{r}\right)\equiv \frac{I\left(\overrightarrow{r}\right)}{{I}_{0}}\left(\overrightarrow{r}\right)-1\approx 2\mathit{Re}\left\{{E}_{S}\left(\overrightarrow{r},0\right)\right\}$$
(4)
$${E}_{S}\left(\overrightarrow{r},z\right)={E}_{S}\left(\overrightarrow{r},0\right)\otimes h\left(\overrightarrow{r},-z\right)$$
(5)
$${E}_{S}\left(\overrightarrow{r},z\right)\approx \frac{{e}^{-ikz}}{4{\pi}^{2}}\underset{-\infty}{\overset{\infty}{\int}}B\left(\overrightarrow{q}\right)H\left(\overrightarrow{q},-z\right){e}^{i\overrightarrow{q}\cdot \overrightarrow{r}}{d}^{2}\overrightarrow{q}$$
(6)
$${\tilde{I}}_{D}\left(\rho \right)=\frac{{\tilde{I}}_{R}\left(\rho \right)}{\tilde{K}\left(\rho \right)+\chi}$$
(7)
$${\overrightarrow{E}}_{s}\left(\overrightarrow{r},0\right)={E}_{0}\left({\overrightarrow{r}}_{p}\right){\overrightarrow{f}}_{s}\left(k\left(\overrightarrow{r}-{\overrightarrow{r}}_{p}\right)\right)$$
(8)
$${\overrightarrow{f}}_{s}\left(k\overrightarrow{r}\right)=\sum _{n=1}^{{n}_{c}}\frac{{i}^{n}\left(2n+1\right)}{n\left(n+1\right)}\left[i{a}_{n}{\overrightarrow{N}}_{e1n}^{\left(3\right)}\left(k\overrightarrow{r}\right)-{b}_{n}{\overrightarrow{M}}_{o1n}^{\left(3\right)}\left(k\overrightarrow{r}\right)\right]$$
(9)
$$B\left(\overrightarrow{r}\right)\equiv \frac{I\left(\overrightarrow{r}\right)}{{I}_{0}\left(\overrightarrow{r}\right)}\approx 1+2\mathit{Re}\left\{{\overrightarrow{f}}_{s}\left(k\left(\overrightarrow{r}-{\overrightarrow{r}}_{p}\right)\right)\cdot {\widehat{\epsilon}}_{0}\right\}+{\left|{\overrightarrow{f}}_{s}\left(k\left(\overrightarrow{r}-{\overrightarrow{r}}_{p}\right)\right)\right|}^{2}$$
(10)
$$\begin{array}{lll}\mathit{Pr}\left(z\right)\hfill & =\hfill & {e}^{\frac{{E}_{\mathit{grav}}+{E}_{\mathit{elect}}}{{K}_{B}T}}\hfill \\ \frac{{E}_{\mathit{grav}}+{E}_{\mathit{elect}}}{{K}_{B}T}\hfill & =\hfill & -\frac{4\pi {a}_{p}^{3}\mathrm{\Delta}\rho g}{3{k}_{B}T}z-\frac{{Q}^{2}{e}^{-\kappa}}{\epsilon {\kappa}^{2}{a}_{p}^{3}}\left(z-{a}_{p}\right)\hfill \end{array}$$