## Abstract

We present a simple-to-align, highly-portable interferometer, which is able to capture wide-field, off-axis interference patterns from transparent samples under low-coherence illumination. This small-dimensions and low-cost device can be connected to the output of a transmission microscope illuminated by a low-coherence source and measure sub-nanometric optical thickness changes in a label-free manner. In contrast to our previously published design, the $\tau $ interferometer, the new design is able to fully operate in an off-axis holographic geometry, where the interference fringes have high spatial frequency, and the interference area is limited only by the coherence length of the source, and thus it enables to easily obtain high-quality quantitative images of static and dynamic samples. We present several applications for the new design including nondestructive optical testing of transparent microscopic elements with nanometric thickness and live-cell imaging.

©2013 Optical Society of America

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### Equations (11)

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(1)
$$\theta =\mathrm{arc}\mathrm{tan}(\Delta y/f),$$
(2)
$$I=\u3008|{V}_{s}+{V}_{r}{|}^{2}\u3009={I}_{s}+{I}_{r}+{G}_{+1}+{G}_{-1}.$$
(3)
$${G}_{+1}=\sqrt{{I}_{s}{I}_{r}}\times \mathrm{exp}\left[-\frac{\left|OP{D}_{total}\right|}{{l}_{c}(x,y)}\right]\times \mathrm{exp}\left[-j\frac{2\pi}{\lambda}OP{D}_{total}\right]\times \mathrm{exp}\left[-j\frac{2\pi}{\lambda}y\mathrm{sin}(\theta )\right],$$
(4)
$$OP{D}_{s}(x,y)=[{\overline{n}}_{s}(x,y)-{n}_{m}]\times {h}_{s}(x,y),$$
(5)
$${\overline{n}}_{s}(x,y)=\frac{1}{{h}_{s}(x,y)}{\displaystyle \underset{0}{\overset{{h}_{s}(x,y)}{\int}}{n}_{s}(x,y,z)dz}.$$
(A1)
$${G}_{+1}=\u3008{V}_{s}^{*}(t)\times {V}_{r}(t+\tau )\u3009=\sqrt{{I}_{s}{I}_{r}}\times \mathrm{exp}\left[-\frac{\left|\tau \right|}{{\tau}_{c}}\right]\times \mathrm{exp}\left[-j2\pi \frac{c}{\lambda}\tau \right],$$
(A2)
$$\begin{array}{c}{t}_{1}=\frac{d}{c};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}_{2}=\frac{1}{c}\left\{[d-{h}_{m}(x,y)]+[{h}_{m}(x,y)-{h}_{s}(x,y)]\times {n}_{m}(x,y)+{h}_{s}(x,y)\times {\overline{n}}_{s}(x,y)\right\};\\ \tau ={t}_{2}-{t}_{1}=\frac{1}{c}\left\{{h}_{s}(x,y)\times [{\overline{n}}_{s}(x,y)-{n}_{m}(x,y)]+{h}_{m}(x,y)\times [{n}_{m}(x,y)-1]\right\},\end{array}$$
(A3)
$$\begin{array}{l}OP{D}_{total}(x,y)=c\cdot \tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={h}_{s}(x,y)\times [{\overline{n}}_{s}(x,y)-{n}_{m}(x,y)]+{h}_{m}(x,y)\times [{n}_{m}(x,y)-1]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=OP{D}_{s}+OP{D}_{m}\text{\hspace{0.17em}},\end{array}$$
(A4)
$$\begin{array}{l}{G}_{+1}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{{I}_{s}{I}_{r}}\times \mathrm{exp}\left[-\left|\frac{OP{D}_{total}}{c}\right|\frac{c}{{l}_{c}}\right]\times \mathrm{exp}\left[-j2\pi \frac{c}{\lambda}\frac{OP{D}_{total}}{c}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sqrt{{I}_{s}{I}_{r}}\times \mathrm{exp}\left[-\left|\frac{OP{D}_{total}}{{l}_{c}}\right|\right]\times \mathrm{exp}\left[-j\frac{2\pi}{\lambda}OP{D}_{total}\right].\end{array}$$
(A5)
$${G}_{+1}=\sqrt{{I}_{r}{I}_{s}}\mathrm{exp}\left[-\frac{\left|OP{D}_{total}\right|}{{l}_{c}(x,y)}\right]\times \mathrm{exp}\left[-j\frac{2\pi}{\lambda}OP{D}_{total}\right]\times \mathrm{exp}\left[-j\frac{2\pi}{\lambda}y\mathrm{sin}(\theta )\right],$$