Transmission Microscopy with Nanometer Resolution Using a Deterministic Single Ion Source

We realize a single particle microscope by using deterministically extracted laser-cooled ${}^{40}{\mathrm{Ca}}^{+}$ ions from a Paul trap as probe particles for transmission imaging. We demonstrate focusing of the ions to a spot size of $5.8\pm 1.0\mathrm{nm}$ and a minimum two-sample deviation of the beam position of 1.5 nm in the focal plane. The deterministic source, even when used in combination with an imperfect detector, gives rise to a fivefold increase in the signal-to-noise ratio as compared with conventional Poissonian sources. Gating of the detector signal by the extraction event suppresses dark counts by 6 orders of magnitude. We implement a Bayes experimental design approach to microscopy in order to maximize the gain in spatial information. We demonstrate this method by determining the position of a $1\mu \mathrm{m}$ circular hole structure to a precision of 2.7 nm using only 579 probe particles.

Figure 1

(a) Sketch of the single ion microscope. (b) Number of detected ions out of 30 single ion extractions at the corresponding profiling edge position (circles). Errors are determined from binomial counting statistics. Dashed line shows center position and gray lines show $1\sigma$ radius of the beam waist. Fitting a Gaussian error function $p(x)=(a/2)[1+\mathrm{erf}\mathbf{(}(x-x_0)/\sigma \sqrt{2}\mathbf{)}]$ to the data yields $a=28.1\pm 0.5$ and $\sigma =5.8\pm 1.0\mathrm{nm}$. (c) Log-log plot of the two-sample deviation of the beam position ${\sigma }_{\mathrm{pos}}$ versus the number of consecutive profiling edge measurements $n$. The fit (black line) to the beam position deviations (circles) reveals a slope of $-0.48\pm 0.01$. Integration time for $n=10$ is about 5 min; for $n=100$ it is about 1 h.

Figure 2

(a) SEM image of the waveguide-cavity structure. Holes have a diameter of about 150 nm. (b) Scan of the cavity structure using one ion at each lateral position, with a resolution of $(25\times 25){\mathrm{nm}}^2$ per pixel. The entire information in the picture is based on 2659 transmission events out of 4141 extracted ions. (c) Imaging a source with emulated Poissonian behavior: The lower SNR as compared to (c) is clearly visible. Missing holes compared to (a) are attributed to blind holes. Here, image information is based on 2420 transmission events out of 3694 extracted ions.

Figure 3

Histogram of optimal blade positions. These data from 500 events are split into cases where the ion was detected (blue) and cases where it was not (purple). Bayes fit function $p(y=1|\theta ,\xi )$ is shown according to the final parameter values (black), radius $\sigma =7.18\pm 0.89\mathrm{nm}$, and detector efficiency $a=0.95\pm 0.02$. The zero of the $x$ axis is set to $x_0$. For comparison, the result of a maximum likelihood fit (dashed red) to the entire data, with $\sigma =7.13\pm 0.66\mathrm{nm}$ and $a=0.95\pm 0.02$, $x_0=0.73\mathrm{nm}$. The inset shows the average deviation of the simulation results from the real value as a function of the number of iterations, using 1000 independent simulation runs for each data point. A multiplicative speed-up of a factor of $\approx 4/3$ is found in determining the beam position and an exponential speed-up from $n^{-0.5}$ to $n^{-0.76}$ in determining the radius when using the Bayesian method.

Figure 4

(a) A circular hole structure is scanned using one ion at each lateral position, with a resolution of $100\times 100{\mathrm{nm}}^2$ per pixel. The red circle shows the result of a maximum likelihood fit to the data. (b) The same structure is measured using the Bayes experimental design method. For the plot, the $x$ and $y$ positions of the hole were set to zero. The blue and red dots represent the positions where an ion was, or was not, detected, respectively. The final location and radius of the hole structure is depicted by the dashed circle. The initial guess of the Gaussian-shaped PDF for the position is depicted as a gray shade in the background. The dark gray line follows the progression of its mean value, i.e., the assumed center position of this distribution as a function of the number of extracted ions. Within the first four iterations, no ion is transmitted. The spatial information of these blocked particles shifts the assumed position, since it excludes that specific areas are transmissive. After the first ion is transmitted, the assumed position makes a step towards its location. (c) shows a histogram of detected and not detected events dependent on $r$ the distance to the center of the structure.