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Optimal configurations of Scanning Helium Microscopes
The optimal configurations of scanning helium microscopes are geometrical configurations that maximise the intensity of the imaging beam within a given lateral resolution and under certain technological constraints.

When designing a scanning helium microscope, scientists strive to maximise the intensity of the imaging beam while minimising its width. The reason behind this is that the beam's width gives the resolution of the microscope while its intensity is proportional to its signal to noise ratio. Due to their neutrality and high ionisation energy, neutral helium atoms are hard to detect. This makes high-intensity beams a crucial requirement for a viable scanning helium microscope.

In order to generate a high-intensity beam, scanning helium microscopes are designed to generate a supersonic expansion of the gas into vacuum, that accelerates neutral helium atoms to high velocities. Scanning helium microscopes exist in two different configurations: the pinhole configuration and the zone plate configuration. In the pinhole configuration, a small opening (the pinhole) selects a section of the supersonic expansion far away from its origin, which has previously been collimated by a skimmer (essentially, another small pinhole). This section then becomes the imaging beam. In the zone plate configuration a fresnel zone plate focuses the atoms coming from a skimmer into a small focal spot.

Each of these configurations have different optimal designs, as they are defined by different optics equations.

Pinhole configuration
For the pinhole configuration the width of the beam (which we aim to minimise) is largely given by geometrical optics. The size of the beam at the sample plane is given by the lines connecting the skimmer edges with the pinhole edges. When the fresnel number is very small ($$F\ll1$$), the beam width is also affected by Fraunhofer diffraction (see equation below).

$$\Phi = 2\sqrt{2\ln 2/3}\sqrt{\delta^2+3\sigma_A^2(1-\theta(F))}.$$

In this equation $$\Phi$$ is the Full Width at Half Maximum of the beam, $$\delta$$ is the geometrical projection of the beam and $$\sigma_A$$ is the Airy diffraction term. $$\theta$$ is the Heaviside step function used here to indicate that the presence of the diffraction term depends on the value of the Fresnel number. Note that there are variations of this equation depending on what is defined as the "beam width" (for details compare and ). Due to the small wavelength of the helium beam, the Fraunhofer diffraction term can usually be omitted.

The intensity of the beam (which we aim to maximise) is given by the following equation (according to the Sikora and Andersen model) :

$$I = I_0 \frac{r_{ph}^2}{(R_{F}+a)^2}\left(1-\exp\left[-S^2\left(\frac{r_S(R_F+a)}{R_F(R_F-x_S+a)}\right) ^2\right]\right).$$

Where $$I_0$$ is the total intensity stemming from the supersonic expansion nozzle (taken as a constant in the optimisation problem), $$r_{ph}$$ is the radius of the pinhole, S is the speed ratio of the beam, $$r_S$$ is the radius of the skimmer, $$R_F$$ is the radius of the supersonic expansion quitting surface (the point in the expansion from which atoms can be considered to travel in a straight line), $$x_S$$is the distance between the nozzle and the skimmer and $$a$$ is the distance between the skimmer and the pinhole. There are several other versions of this equation that depend on the intensity model, but they all show a quadratic dependency on the pinhole radius (the bigger the pinhole, the more intensity) and an inverse quadratic dependency with the distance between the skimmer and the pinhole (the more the atoms spread, the less intensity).

By combining the two equations shown above, one can obtain that for a given beam width $$\Phi$$ for the geometrical optics regime the following values correspond to intensity maxima:

$$r_S^{max}=\frac{\Phi a}{2W_D K}, \qquad r_{ph}^{max}=\frac{\Phi a}{2K(a+W_D)}.$$

In here, $$W_D$$ represents the working distance of the microscope and $$K=2\sqrt{2\ln 2/3}$$ is a constant that stems from the definition of the beam width. Note that both equations are given with respect to the distance between the skimmer and the pinhole, a. The global maximum of intensity can then be obtained numerically by replacing these values in the intensity equation above. In general, smaller skimmer radii coupled with smaller distances between the skimmer and the pinhole are preferred, leading in practice to the design of increasingly smaller pinhole microscopes.

Zone plate configuration
The zone plate microscope uses a zone plate (that acts roughly like as a classical lens) instead of a pinhole to focus the atom beam into a small focal spot. This means that the beam width equation changes significantly (see below).

$$\Phi = K\sqrt{\sigma_{cm}^2+\sigma_A^2+\left(\frac{M r_S}{\sqrt{3}}\right)^2} \sim K\sqrt{\left(\frac{r_{ZP}}{S\sqrt{2}}\right)^2+0.42\Delta r+\left(\frac{M r_S}{\sqrt{3}}\right)^2 }.$$

Here $$M$$ is the zone plate magnification and $$\Delta r$$ is the width of the smallest zone. Note the presence of chromatic aberrations ($$\sigma_{cm}$$). The approximation sign indicates the regime in which the distance between the zone plate and the skimmer is much bigger than its focal length.

The first term in this equation is similar to the geometric contribution $$\delta$$ in the pinhole case: a bigger zone plate (taken all parameters constant) corresponds to a bigger focal spot size. The third term differs from the pinhole configuration optics as it includes a quadratic relation with the skimmer size (which is imaged through the zone plate) and a linear relation with the zone plate magnification, which will at the same time depend on its radius.

The equation to maximise, the intensity, is the same as the pinhole case with the substitution $$r_{ph}\leftrightarrow r_{ZP}$$. By substitution of the magnification equation:

$$M=\frac{f}{a-f}=\frac{2r_{ZP}\Delta r}{\lambda(a-2r_{ZP}\Delta r/\lambda)}.$$

Where $$\lambda$$ is the average de-Broglie wavelength of the beam. Taking a constant $$\Delta r$$, which should be made equal to the smallest achievable value, the maxima of the intensity equation with respect to the zone plate radius and the skimmer-zone plate distance $$a$$ can be obtained analytically. The derivative of the intensity with respect to the zone plate radius can be reduced the following cubic equation (once it has been set equal to zero):

$$a^3+2a^2\left(R_F-\sqrt{3\Gamma}r_{ZP}\right)+a R_F(R_F-4r_{ZP}\sqrt{3\Gamma}) = r_{ZP}\sqrt{3\Gamma})R_F^2\left[\frac{2S^2\Phi'^2+r_{ZP}^2(\Gamma-1)}{S^2\Phi'^2-0.5r_{ZP}^2}\right].$$

Here some groupings are used: $$\Gamma$$ is a constant that gives the relative size of the smallest aperture of the zone plate compared with the average wavelength of the beam and $$\Phi'$$ is the modified beam width, which is used through the derivation to avoid explicitly operating with the constant airy term: $$\Phi'^2=\sigma_{cm}^2+\left( \frac{M r_S}{\sqrt{3}}\right)^2$$.

This cubic equation is obtained under a series of geometrical assumptions and has a closed-form analytical solution that can be consulted in the original paper or obtained through any modern-day algebra software. The practical consequence of this equation is that zone plate microscopes are optimally designed when the distances between the components are small, and the radius of the zone plate is also small. This goes in line with the results obtained for the pinhole configuration, and has as its practical consequence the design of smaller scanning helium microscopes.