Atomic scale displacements detected by optical image cross-correlation analysis and 3D printed marker arrays

Abstract

For analyzing displacement-vector fields in mechanics, for example to characterize the properties of 3D printed mechanical metamaterials, routine high-precision position measurements are indispensable. For this purpose, nanometer-scale localization errors have been achieved by wide-field optical-image cross-correlation analysis. Here, we bring this approach to atomic-scale accuracy by combining it with well-defined 3D printed marker arrays. By using an air-lens with a numerical aperture of [Formula: see text] and a free working distance of [Formula: see text], and an [Formula: see text] array of markers with a diameter of [Formula: see text] and a period of [Formula: see text], we obtain 2D localization errors as small as [Formula: see text] in [Formula: see text] measurement time ([Formula: see text]). The underlying experimental setup is simple, reliable, and inexpensive, and the marker arrays can easily be integrated onto and into complex architectures during their 3D printing process.

Figure 1

Scheme of the simple optical setup used to determine two-dimensional displacement vectors of a macroscopic sample with atomic-scale localization errors. The surface of a sample is illuminated by unpolarized visible white light from a filtered incandescent source impinging onto the sample under an angle. An objective lens (with focal length $f=8.25\mathrm{mm}$) together with a tube lens (with focal length $f=200\mathrm{mm}$) images the sample surface onto a digital black/white camera. The objective lens has a numerical aperture of $\mathrm{NA}=0.4$ and a free working distance of $11.2\mathrm{mm}$. The images acquired by the camera are processed using image cross-correlation analysis. We can displace the sample in the plane normal to the optical axis by a precision piezoelectric stage. The setup is located on a vibration-isolated optical table and enclosed in a box to reduce vibrations and drifts between the sample and the camera position.

Figure 2

Top-view electron micrographs of four of the five investigated samples. (a) Sample #1 is a sandblasted copper surface. (b) Sample #2 is a glass substrate with randomly distributed micrometer-sized gold grains on top. Sample #3 (not depicted) is a glass substrate with a square array of polymer markers with period $a=10\mathrm{\mu }\mathrm{m}$ on top, fabricated by 3D laser printing. Without metal coating, this sample cannot easily be imaged by electron microscopy. (c) Sample #4 is as sample #3, but coated with a $54\mathrm{nm}$ thin film of gold. (d) Sample #5 is as sample #4, but with a period of $a=5\mathrm{\mu }\mathrm{m}$.

Figure 3

Summary of data obtained from five different samples #1 to #5 (cf. Fig. 2). Column (a) exhibits an example optical image with the used regions of interest (ROI) indicated by the blue squares. Each ROI comprises $30\times 30$ camera pixels. The ROI lie in a footprint of ${\left(40,,\mathrm{\mu },\mathrm{m}\right)}^2$ indicated by the dashed white square. Column (b) shows results obtained from the optical-image cross-correlation approach for the $x$-component (red) and the $y$-component (blue). For comparison, the read-out signal from the capacitive sensor of the piezoelectric actuator is shown in gray. This signal has been shifted vertically for clarity. For each of the $800$ data points, we obtain localization errors ${\sigma }_x$ and ${\sigma }_y$. The mean values $⟨{\sigma }_x⟩$ and $⟨{\sigma }_x⟩$ over $800$ measurements are indicated. $⟨{\sigma }_x^{{}^{\prime }}⟩$ is the corresponding value for the capacitive sensor, for the same measurement time of $12.5\mathrm{ms}$. In column (b), the piezoelectric actuator has not been moved intentionally. In contrast, in column (c), the piezoelectric actuator has been moved in a staircase manner with $1\mathrm{nm}$ high steps each $0.5\mathrm{s}$.